Analyse des ventes du site web Lapage

Importation des données

clients <- read.csv("customers.csv", header = TRUE)
produits <- read.csv("products.csv", header = TRUE)
transactions <- read.csv("transactions.csv", header = TRUE)

Vérification de la présence d’éventuels outliers

summary(clients)

##   client_id             sex                birth     
##  Length:8623        Length:8623        Min.   :1929  
##  Class :character   Class :character   1st Qu.:1966  
##  Mode  :character   Mode  :character   Median :1979  
##                                        Mean   :1978  
##                                        3rd Qu.:1992  
##                                        Max.   :2004

summary(produits)

##    id_prod              price            categ       
##  Length:3287        Min.   : -1.00   Min.   :0.0000  
##  Class :character   1st Qu.:  6.99   1st Qu.:0.0000  
##  Mode  :character   Median : 13.06   Median :0.0000  
##                     Mean   : 21.86   Mean   :0.3702  
##                     3rd Qu.: 22.99   3rd Qu.:1.0000  
##                     Max.   :300.00   Max.   :2.0000

summary(transactions)

##    id_prod              date            session_id         client_id        
##  Length:679532      Length:679532      Length:679532      Length:679532     
##  Class :character   Class :character   Class :character   Class :character  
##  Mode  :character   Mode  :character   Mode  :character   Mode  :character

Nous observons qu’il y a au moins un prix négatif au sein du dataframe “produits”. Nous prendrons le parti de le(s) remplacer par la moyenne à savoir 21,86. Nous allons également transposer les catégories en facteurs.

produits$categ <- as.factor(produits$categ)

Suppression des prix négatifs

Commençons par créer une copie de “produits”, “clients” et “transactions :

prod <- produits
clients2 <- clients
trans <- transactions

Remplaçons maintenant les valeurs négatives par la moyenne :

prod$price[prod$price<=0] <- 21.86

Assurons nous qu’il n’y ait plus de prix négatif :

summary(prod)

##    id_prod              price        categ   
##  Length:3287        Min.   :  0.62   0:2309  
##  Class :character   1st Qu.:  6.99   1: 739  
##  Mode  :character   Median : 13.09   2: 239  
##                     Mean   : 21.86           
##                     3rd Qu.: 22.99           
##                     Max.   :300.00

Tout est ok ✓.

Vérification de la présence d’id aberrants

outlier_prod <- unique(prod$id_prod)
outlier_client <- unique(clients2$client_id)

En observant le dataframe “transactions”, nous nous apercevons de la présence de transactions test, toutes effectuées à la même date (2021-03-01) par des clients fictifs (ct_0 et ct_1) sur le même produit (T_0).

head(sort(outlier_prod, decreasing = TRUE))

## [1] "T_0"  "2_99" "2_98" "2_97" "2_96" "2_95"

head(sort(outlier_client, decreasing = TRUE))

## [1] "ct_1"  "ct_0"  "c_999" "c_998" "c_997" "c_996"

Suppression des id aberrants

Vérifions le prix du produit correspondant à l’id T_0 :

prod$price[prod$id_prod == "T_0"]

## [1] 21.86

Au vue de la structure de leurs id, il semblerait qu’il s’agisse de transactions tests réalisées probablement lors du lancement du site. Nous décidons donc de supprimer le produit, les clients ainsi que les transactions associées à ces derniers :

prod_t <- which(prod$id_prod == "T_0")
prod <- prod[-prod_t,]

clients_t0 <- which(clients2$client_id == "ct_0")
clients_t1 <- which(clients2$client_id == "ct_1")
clients2 <- clients2[-clients_t1,]
clients2 <- clients2[-clients_t0,]

trans_t <- which(trans$id_prod == "T_0")
trans <- trans[-trans_t,]

Tout a bien été supprimé ✓.

Vérification de la présence de clients inactifs et analyse de ces derniers

Isolement des clients inactifs

id_clients1 <- unique(clients2$client_id)
id_clients2 <- unique(trans$client_id) 
id_clients3 <- intersect(id_clients1, id_clients2)

Nous avons un total de 21 clients inactifs au sein de notre base de données. Isolons les afin d’en analyser, par la suite, les profils.

clients_actifs <- which(is.element(clients2$client_id, id_clients3))
clients_i <- clients2[-clients_actifs,]
age <- 2023-clients_i$birth
clients_i <- data.frame(clients_i, age)
head(clients_i)

##      client_id sex birth age
## 802     c_8253   f  2001  22
## 2484    c_3789   f  1997  26
## 2735    c_4406   f  1998  25
## 2770    c_2706   f  1967  56
## 2852    c_3443   m  1959  64
## 3180    c_4447   m  1956  67

Profils des clients inactifs

library(magrittr)
library(dplyr)

## 
## Attaching package: 'dplyr'

## The following objects are masked from 'package:stats':
## 
##     filter, lag

## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, setequal, union

clients_inactifs <- clients_i %>% group_by(sex) %>% summarise(nb = n(), prop = round(nb/21*100, digits = 2), age_moyen = round(sum(age)/nb, digits = 2))
clients_inactifs_age <- clients_i %>% group_by(age) %>% summarise(nb = n(), prop = round(nb/21*100, digits = 2))
head(clients_inactifs)

## # A tibble: 2 × 4
##   sex      nb  prop age_moyen
##   <chr> <int> <dbl>     <dbl>
## 1 f        11  52.4      38.1
## 2 m        10  47.6      39.2

head(clients_inactifs_age)

## # A tibble: 6 × 3
##     age    nb  prop
##   <dbl> <int> <dbl>
## 1    19     3 14.3 
## 2    20     1  4.76
## 3    21     1  4.76
## 4    22     2  9.52
## 5    24     1  4.76
## 6    25     1  4.76

library(ggplot2)
library(hrbrthemes)

## NOTE: Either Arial Narrow or Roboto Condensed fonts are required to use these themes.

##       Please use hrbrthemes::import_roboto_condensed() to install Roboto Condensed and

##       if Arial Narrow is not on your system, please see https://bit.ly/arialnarrow

ggplot(clients_inactifs, aes(x=sex, y=age_moyen)) + 
  geom_bar(stat = "identity", width=0.5) +
  ylab("Âge moyen") +
  xlab("Sexe") +
  ggtitle("Âge moyen des clients inactifs en fonction du sexe") +
  theme_ipsum() +
  coord_flip()

library(ggplot2)
library(hrbrthemes)

ggplot(clients_inactifs_age, aes(x=nb, y=age, fill=nb)) +
  ggtitle("Répartition des clients inactifs en fonction de leur âge") +
  xlab("Nombre de personnes") +
  ylab("Âge des clients") +
  theme_ipsum() +
  geom_violin()

library(ggplot2)

clients_inactifs$ymax <- cumsum(clients_inactifs$prop)
clients_inactifs$ymin <- c(0, head(clients_inactifs$ymax, n=-1))
clients_inactifs$labelPosition <- (clients_inactifs$ymax + clients_inactifs$ymin) / 2
clients_inactifs$label <- paste0(clients_inactifs$sex, "\n proportion(en %): \n", clients_inactifs$prop)
ggplot(clients_inactifs, aes(ymax=ymax, ymin=ymin, xmax=4, xmin=3, fill=sex)) +
  geom_rect() +
  geom_label( x=3.5, aes(y=labelPosition, label=label), size=6) +
  scale_fill_brewer(palette=4) +
  coord_polar(theta="y") +
  xlim(c(2, 4)) +
  ggtitle("Répartition des clients inactifs en fonction du sexe") +
  theme_void() +
  theme(legend.position = "none")

Nous pouvons maintenant continuer en n’utilisant que les clients actifs :

clients2 <- clients2[is.element (clients2$client_id, id_clients3),]
head(clients2)

##   client_id sex birth
## 1    c_4410   f  1967
## 2    c_7839   f  1975
## 3    c_1699   f  1984
## 4    c_5961   f  1962
## 5    c_5320   m  1943
## 6     c_415   m  1993

Vérification de la présence de produits invendus et analyse de ces derniers

Isolement des produits jamais vendus

id_prod1 <- unique(prod$id_prod)
id_prod2 <- unique(trans$id_prod) 
id_prod3 <- intersect(id_prod1, id_prod2)

21 références présentes dans la table des produits n’ont jamais été achetées. Isolons-les également afin d’en analyser, par la suite, les caractéristiques :

ref_vendues <- which(is.element(prod$id_prod, id_prod3))
ref_invendues <- prod[-ref_vendues,]
head(ref_invendues)

##     id_prod price categ
## 185  0_1016 35.06     0
## 280  0_1780  1.67     0
## 738  0_1062 20.08     0
## 795  0_1119  2.99     0
## 812  0_1014  1.15     0
## 847     1_0 31.82     1

Caractéristiques des références invendues

Commençons par classer les références invendues en fonction de leurs catégories :

library(magrittr)
library(dplyr)

ref_invendues_categ <- ref_invendues %>% group_by(categ) %>% summarise(nb = n(), prop = round(nb/21*100, digits = 2), prix_moyen = round(sum(price)/nb, digits = 2))
head(ref_invendues_categ)

## # A tibble: 3 × 4
##   categ    nb  prop prix_moyen
##   <fct> <int> <dbl>      <dbl>
## 1 0        16 76.2        14.3
## 2 1         2  9.52       35.8
## 3 2         3 14.3       165.

Représentons graphiquement ce dataframe afin d’en faciliter la compréhension :

library(ggplot2)
library(hrbrthemes)

ggplot(ref_invendues_categ, aes(fill=categ, y=prop, x="Références")) + 
    geom_bar(position="fill", stat="identity") +
    ylab("Proportion") +
    theme_ipsum() +
    ggtitle("Répartition des catégories des produits invendus")

La catégorie 0 est la plus largement représentée dans les références invendues. Intéressons nous maintenant aux prix des références invendues :

library(magrittr)
library(dplyr)

ref_invendues_price <- ref_invendues %>% group_by(categ) %>% summarise(nb = n(), prop = round(nb/21*100, digits = 2), prix_moyen = round(sum(price)/nb, digits = 2))
head(ref_invendues_price)

## # A tibble: 3 × 4
##   categ    nb  prop prix_moyen
##   <fct> <int> <dbl>      <dbl>
## 1 0        16 76.2        14.3
## 2 1         2  9.52       35.8
## 3 2         3 14.3       165.

Représentons graphiquement la répartition des prix des références invendues :

library(ggplot2)

ref_invendues %>%
  ggplot( aes(x=price)) +
    geom_density(fill="#69b3a2", color="#e9ecef", alpha=0.8) +
    xlab("Prix") +
    ylab("Densité") +
    theme_ipsum() +
    ggtitle("Répartition des prix des références invendues")

Et avec une toute autre représentation :

library(ggplot2)

ggplot(ref_invendues, aes(x="Références", y=price)) + 
    geom_boxplot(fill="slateblue", alpha=0.5) +
    ylab("Prix") +
    theme_ipsum() +
    ggtitle("Répartition des prix des références invendues")

La catégorie 0 étant la plus représentée dans cet échantillon, il n’est pas étonnant que la grosse majorité des prix soit inférieure à 50€ (la catégorie 0 ayant le prix moyen le plus faible). Qu’en est-il de la répartition des prix par catégorie ?

library(ggplot2)

ggplot(ref_invendues, aes(x=as.factor(categ), y=price)) + 
    geom_boxplot(fill="slateblue", alpha=0.2) + 
    xlab("Catégorie") +
    ylab("Prix") +
    theme_ipsum() +
    ggtitle("Répartition des prix par catégorie")

Poursuivons maintenant en utilisant uniquement les références vendues :

prod <- prod[is.element (prod$id_prod, id_prod3),]

Vérification de la présence de doublons

doublons_c <- sum(duplicated(clients2$client_id))
doublons_p <- sum(duplicated(prod$id_prod))
doublons_t <- sum(duplicated(trans$session_id))

Nous n’avons pas de doublons dans les dataframes “prod” et “clients”. En ce qui concerne les transactions, nous n’avons pas de clé primaire relative à celles-ci et il est raisonnable d’avoir des doublons au niveau des session_id dans la mesure où plusieurs transactions peuvent intervenir lors d’une même session.

Vérification de la présence de NA

na_p <- which(is.na(prod))
na_c <- which(is.na(clients2))
na_t <- which(is.na(trans))
na_p

## integer(0)

na_c

## integer(0)

na_t

## integer(0)

Nous n’avons pas de NA ✓.

Création du dataframe global

Commençons par vérifier la classe des dates que nous avons :

str(trans$date)

##  chr [1:679332] "2022-05-20 13:21:29.043970" "2022-02-02 07:55:19.149409" ...

Convertissons la chaîne de charactère en date afin de faciliter la manipulation de celles-ci :

trans$date <- as.Date(trans$date)

Assurons nous que le changement a bien été effectué :

str(trans$date)

##  Date[1:679332], format: "2022-05-20" "2022-02-02" "2022-06-18" "2021-06-24" "2023-01-11" ...

Changement opéré ✓.

Jointure entre les transactions et les produits

df1 <- merge(trans, prod, all = T)
head(df1)

##   id_prod       date session_id client_id price categ
## 1     0_0 2022-02-22   s_169315    c_1494  3.75     0
## 2     0_0 2021-10-29   s_111848    c_2068  3.75     0
## 3     0_0 2022-06-22   s_227677    c_3454  3.75     0
## 4     0_0 2022-10-28   s_289299    c_1126  3.75     0
## 5     0_0 2022-11-22   s_301230     c_803  3.75     0
## 6     0_0 2022-02-08   s_162168    c_3172  3.75     0

Nous avons opté pour une jointure externe afin de conserver l’intégralité des lignes de nos dataframes. Vérifions maintenant si nous avons des NAs dans ce nouveau dataframe, et si oui, où ils se trouvent :

summary(df1)

##    id_prod               date             session_id         client_id        
##  Length:679332      Min.   :2021-03-01   Length:679332      Length:679332     
##  Class :character   1st Qu.:2021-09-08   Class :character   Class :character  
##  Mode  :character   Median :2022-03-03   Mode  :character   Mode  :character  
##                     Mean   :2022-03-03                                        
##                     3rd Qu.:2022-08-30                                        
##                     Max.   :2023-02-28                                        
##                                                                               
##      price         categ       
##  Min.   :  0.62   0   :415459  
##  1st Qu.:  8.87   1   :227169  
##  Median : 13.99   2   : 36483  
##  Mean   : 17.45   NA's:   221  
##  3rd Qu.: 18.99                
##  Max.   :300.00                
##  NA's   :221

Nous avons des données manquantes au niveau des prix et des catégories. Étant donné que nous avions préalablement vérifié s’il y avait des NAs dans nos différentes tables, nous pouvons supposer que la table des transactions contenait un id_prod qui n’est pas présent dans la table des produits et qui s’est retrouvé dans df1 suite à la jointure. Vérifions cette hypothèse :

id_prod21 <- unique(df1$id_prod)
id_prod22 <- unique(prod$id_prod) 
id_prod23 <- intersect(id_prod1, id_prod2)
non_na <- which(is.element(df1$id_prod, id_prod23))
na <- df1[-non_na,]

Nous avons effectivement un id_produit présent dans df1 mais pas dans la table des produits et au vue de sa syntaxe, nous pouvons conclure qu’il appartient à la catégorie 0. Nous allons donc remplacer les NAs par la catégorie 0 et par le prix moyen constaté pour cette même catégorie :

pm_0 <- mean(produits[(produits$categ == 0),]$price)
pm_0

## [1] 11.72728

df1$price[df1$id_prod=="0_2245"] <- pm_0
df1$categ[df1$id_prod=="0_2245"] <- 0

Vérifions qu’il n’y a plus de NAs au niveau des prix et des catégories :

summary(df1)

##    id_prod               date             session_id         client_id        
##  Length:679332      Min.   :2021-03-01   Length:679332      Length:679332     
##  Class :character   1st Qu.:2021-09-08   Class :character   Class :character  
##  Mode  :character   Median :2022-03-03   Mode  :character   Mode  :character  
##                     Mean   :2022-03-03                                        
##                     3rd Qu.:2022-08-30                                        
##                     Max.   :2023-02-28                                        
##      price        categ     
##  Min.   :  0.62   0:415680  
##  1st Qu.:  8.87   1:227169  
##  Median : 13.99   2: 36483  
##  Mean   : 17.45             
##  3rd Qu.: 18.99             
##  Max.   :300.00

C’est parfait ✓.

Jointure entre df1 et les clients

clients2$age <- 2023-clients2$birth
data <- merge(df1, clients2)
head(data)

##   client_id id_prod       date session_id price categ sex birth age
## 1       c_1  0_1378 2021-08-23    s_79696 13.96     0   m  1955  68
## 2       c_1  0_1090 2021-12-19   s_136532 13.78     0   m  1955  68
## 3       c_1  0_1547 2021-09-08    s_86739  8.99     0   m  1955  68
## 4       c_1   1_713 2021-11-15   s_120172 33.99     1   m  1955  68
## 5       c_1   1_364 2022-06-15   s_224447 10.30     1   m  1955  68
## 6       c_1  0_2277 2021-09-06    s_85977 10.99     0   m  1955  68

Afin de faciliter la manipulation des données, décomposons les dates de notre dataframe :

library(lubridate)

## 
## Attaching package: 'lubridate'

## The following objects are masked from 'package:base':
## 
##     date, intersect, setdiff, union

library(zoo)

## 
## Attaching package: 'zoo'

## The following objects are masked from 'package:base':
## 
##     as.Date, as.Date.numeric

data$date <- ymd(data$date)
data$annee_mois <- data$date
data$annee_mois <- format(data$annee_mois, "%Y-%m")
data$annee_mois <- ym(data$annee_mois)
data$jour <- mday(data$date)
data$mois <- month(data$date)
data$annee <- year(data$date)
head(data)

##   client_id id_prod       date session_id price categ sex birth age annee_mois
## 1       c_1  0_1378 2021-08-23    s_79696 13.96     0   m  1955  68 2021-08-01
## 2       c_1  0_1090 2021-12-19   s_136532 13.78     0   m  1955  68 2021-12-01
## 3       c_1  0_1547 2021-09-08    s_86739  8.99     0   m  1955  68 2021-09-01
## 4       c_1   1_713 2021-11-15   s_120172 33.99     1   m  1955  68 2021-11-01
## 5       c_1   1_364 2022-06-15   s_224447 10.30     1   m  1955  68 2022-06-01
## 6       c_1  0_2277 2021-09-06    s_85977 10.99     0   m  1955  68 2021-09-01
##   jour mois annee
## 1   23    8  2021
## 2   19   12  2021
## 3    8    9  2021
## 4   15   11  2021
## 5   15    6  2022
## 6    6    9  2021

Nous venons d’ajouter des colonnes représentant les éléments qui composent les dates (jours, mois et années). Vérifions une nouvelle fois si nous avons des NAs :

summary(data)

##   client_id           id_prod               date             session_id       
##  Length:679332      Length:679332      Min.   :2021-03-01   Length:679332     
##  Class :character   Class :character   1st Qu.:2021-09-08   Class :character  
##  Mode  :character   Mode  :character   Median :2022-03-03   Mode  :character  
##                                        Mean   :2022-03-03                     
##                                        3rd Qu.:2022-08-30                     
##                                        Max.   :2023-02-28                     
##      price        categ          sex                birth           age       
##  Min.   :  0.62   0:415680   Length:679332      Min.   :1929   Min.   :19.00  
##  1st Qu.:  8.87   1:227169   Class :character   1st Qu.:1970   1st Qu.:36.00  
##  Median : 13.99   2: 36483   Mode  :character   Median :1980   Median :43.00  
##  Mean   : 17.45                                 Mean   :1978   Mean   :45.19  
##  3rd Qu.: 18.99                                 3rd Qu.:1987   3rd Qu.:53.00  
##  Max.   :300.00                                 Max.   :2004   Max.   :94.00  
##    annee_mois              jour            mois            annee     
##  Min.   :2021-03-01   Min.   : 1.00   Min.   : 1.000   Min.   :2021  
##  1st Qu.:2021-09-01   1st Qu.: 8.00   1st Qu.: 3.000   1st Qu.:2021  
##  Median :2022-03-01   Median :16.00   Median : 6.000   Median :2022  
##  Mean   :2022-02-16   Mean   :15.76   Mean   : 6.504   Mean   :2022  
##  3rd Qu.:2022-08-01   3rd Qu.:23.00   3rd Qu.: 9.000   3rd Qu.:2022  
##  Max.   :2023-02-01   Max.   :31.00   Max.   :12.000   Max.   :2023

Rien à signaler.

Calculs autour du chiffre d’affaires

ca_total <- round(sum(data$price), digits = 2)
ca_total

## [1] 11856320

nt_ventes <- length(data$session_id)
nt_ventes

## [1] 679332

panier_moyen <- mean(data$price)
panier_moyen

## [1] 17.45291

Le chiffre d’affaires global du site s’élève à environ 11 856 318€ sur un total de 679 332 transactions, soit un montant moyen de 17,45€ par transaction.

Calcul du chiffre d’affaires par année (calendaire)

library(magrittr)
library(dplyr)

data_per_year_c <- data %>% group_by(annee) %>% summarise(ca = sum(price), nb_ventes = n())
data_per_year_c

## # A tibble: 3 × 3
##   annee       ca nb_ventes
##   <dbl>    <dbl>     <int>
## 1  2021 4771847.    278335
## 2  2022 6110089.    346500
## 3  2023  974384.     54497

2022 étant la seule année civile complète de notre dataframe, il n’est pas surprenant que ce soit l’année ayant généré le plus de chiffre d’affaires et de ventes. Intéressons-nous donc plutôt aux ventes mensuelles.

Calcul du chiffre d’affaires par mois

Nous allons maintenant calculer le chiffre d’affaires classé par mois :

library(magrittr)
library(dplyr)
library(tsibble)

## 
## Attaching package: 'tsibble'

## The following object is masked from 'package:zoo':
## 
##     index

## The following object is masked from 'package:lubridate':
## 
##     interval

## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, union

data_per_month <- unique(data %>% group_by(annee_mois) %>% summarise(annee=year(annee_mois), mois=month(annee_mois), ca = sum(price), nb_ventes = n()))

## `summarise()` has grouped output by 'annee_mois'. You can override using the
## `.groups` argument.

data_per_month$annee_mois <- yearmonth(data_per_month$annee_mois)
head(data_per_month)

## # A tibble: 6 × 5
## # Groups:   annee_mois [6]
##   annee_mois annee  mois      ca nb_ventes
##        <mth> <dbl> <dbl>   <dbl>     <int>
## 1   2021 Mar  2021     3 482546.     28610
## 2   2021 Apr  2021     4 476273.     28457
## 3   2021 May  2021     5 493037.     28293
## 4   2021 Jun  2021     6 484171.     26857
## 5   2021 Jul  2021     7 482882.     24742
## 6   2021 Aug  2021     8 482390.     25659

Représentons graphiquement ce dataframe afin de pouvoir observer l’évolution du chiffre d’affaires au cours des mois :

library(magrittr)
library(ggplot2)
library(hrbrthemes)

data_per_month %>% 
  tail(24) %>% 
  ggplot(aes(x=annee_mois, y=ca, ymin=0, ymax=700000)) +
    geom_line(color="grey") +
    geom_point(shape=21, color="black", fill="#3398FF", size=3) +
    xlab("Mois") +
    ylab("Chiffre d'affaires") +
    theme_ipsum() +
    ggtitle("Évolution mensuelle du chiffre d'affaires")

Jetons également un oeil à l’évolution du montant du panier moyen au cours des mois :

library(magrittr)
library(ggplot2)
library(hrbrthemes)

data_per_month %>% 
  tail(24) %>% 
  ggplot(aes(x=annee_mois, y=ca/nb_ventes, ymin=0, ymax=25)) +
    geom_line(color="grey") +
    geom_point(shape=21, color="black", fill="#3398FF", size=3) +
    xlab("Mois") +
    ylab("Montant du panier moyen") +
    theme_ipsum() +
    ggtitle("Évolution mensuelle du panier moyen")

Nous pouvons observer que le chiffre d’affaires et le montant du panier moyen ont connu une baisse au cours du mois d’octobre 2021. Essayons de comprendre à quoi peuvent être dûes ces importantes diminutions :

library(dplyr)
library(magrittr)

data_per_month_categ <- data %>% group_by(categ, annee_mois) %>% summarise(ca=sum(price), nb_ventes=n())

## `summarise()` has grouped output by 'categ'. You can override using the
## `.groups` argument.

head(data_per_month_categ)

## # A tibble: 6 × 4
## # Groups:   categ [1]
##   categ annee_mois      ca nb_ventes
##   <fct> <date>       <dbl>     <int>
## 1 0     2021-03-01 193735.     18140
## 2 0     2021-04-01 205387.     19356
## 3 0     2021-05-01 196281.     18509
## 4 0     2021-06-01 168025.     15905
## 5 0     2021-07-01 144798.     13582
## 6 0     2021-08-01 167843.     15737

Représentons ces résultats graphiquement afin de mieux les analyser :

library(magrittr)
library(ggplot2)
library(hrbrthemes)

ggplot(data_per_month_categ, aes(fill=categ, y=ca, x=annee_mois)) +
  geom_bar(stat="identity", position=position_dodge2()) +
  xlab("Mois") +
  ylab("Chiffre d'affaires") +
  theme_ipsum() +
  ggtitle("Répartition du chiffre d'affaires mensuel par catégorie")

library(magrittr)
library(ggplot2)
library(hrbrthemes)

ggplot(data_per_month_categ, aes(fill=categ, y=nb_ventes, x=annee_mois)) +
  geom_bar(stat="identity", position=position_dodge2()) +
  xlab("Mois") +
  ylab("Nombre de ventes") +
  theme_ipsum() +
  ggtitle("Répartition des ventes mensuelles par catégorie")

Nous pouvons clairement remarquer que les ventes de livres de la catégorie 1 ont subi une forte baisse au mois d’octobre 2021, baisse qui ne se retrouve pas plus tard. Il doit y avoir un problème au niveau des données qui ont été enregistrées.

Cas du mois d’octobre 2021

Penchons-nous plus particulièrement sur le mois d’octobre 2021 et sur les transactions qui ont été enregistrées sur cette période :

library(magrittr)
library(dplyr)

ventes_oct21 <- data[(data$mois == "10" & data$annee == "2021"),] %>% group_by(categ, jour) %>% summarise(ca=sum(price), nb_ventes=n())

## `summarise()` has grouped output by 'categ'. You can override using the
## `.groups` argument.

head(ventes_oct21)

## # A tibble: 6 × 4
## # Groups:   categ [1]
##   categ  jour    ca nb_ventes
##   <fct> <int> <dbl>     <int>
## 1 0         1 6950.       663
## 2 0         2 7141.       661
## 3 0         3 6787.       648
## 4 0         4 6557.       603
## 5 0         5 6358.       594
## 6 0         6 7547.       702

Représentons ces données graphiquement en groupant par catégorie :

library(magrittr)
library(dplyr)
library(ggplot2)
library(graphics)

par(mfrow=c(3,1))
ventes_oct21 %>%
  filter(categ == "0") %>%
  ggplot(aes(x=jour, y=nb_ventes)) + 
  geom_bar(stat = "identity", width=0.5, color="grey", fill="lightblue") +
  xlab("Jour") +
  ylab("Nombre de ventes") +
  ggtitle("Ventes enregistrées en octobre 2021 (catégorie 0)") +
  theme_ipsum()

ventes_oct21 %>%
  filter(categ == "1") %>%
  ggplot(aes(x=jour, y=nb_ventes)) + 
  geom_bar(stat = "identity", width=0.5, color="grey", fill="blue") +
  xlab("Jour") +
  ylab("Nombre de ventes") +
  ggtitle("Ventes enregistrées en octobre 2021 (catégorie 1)") +
  theme_ipsum()

ventes_oct21 %>%
  filter(categ == "2") %>%
  ggplot(aes(x=jour, y=nb_ventes)) + 
  geom_bar(stat = "identity", width=0.5, color="grey", fill="darkblue") +
  xlab("Jour") +
  ylab("Nombre de ventes") +
  ggtitle("Ventes enregistrées en octobre 2021 (catégorie 2)") +
  theme_ipsum()

Aucune transaction n’a été enregistrée pour les livres de la catégorie 1 entre le 1er et le 28 octobre 2021 (ce qui est probablement dû à un problème d’acquisition des données), d’où la baisse significative de chiffre d’affaires sur ce mois.

Calcul du chiffre d’affaires par catégorie

Commençons par classer nos données par catégorie :

library(dplyr)
library(magrittr)

data_per_cat <- data %>% group_by(categ) %>% summarise(ca = sum(price), prix_moyen = round(mean(price), digits = 2), prop_ca = round(ca/ca_total*100, digits = 2), nb_ventes = n(), prop_ventes = round(nb_ventes/679332*100, digits = 2))
head(data_per_cat)

## # A tibble: 3 × 6
##   categ       ca prix_moyen prop_ca nb_ventes prop_ventes
##   <fct>    <dbl>      <dbl>   <dbl>     <int>       <dbl>
## 1 0     4422323.       10.6    37.3    415680       61.2 
## 2 1     4653723.       20.5    39.2    227169       33.4 
## 3 2     2780275.       76.2    23.4     36483        5.37

Représentons graphiquement les répartitions de chiffres d’affaires et ventes par catégorie :

library(magrittr)
library(ggplot2)
library(hrbrthemes)
library(graphics)

par(mfrow=c(2,1))

data_per_cat %>%
  ggplot(aes(x="", y=prop_ca, fill=categ)) +
  geom_bar(stat="identity", width=1, color="white") +
  coord_polar("y", start=0) +
  theme_void() +
  ggtitle("Chiffre d'affaires par catégorie")

data_per_cat %>%
  ggplot(aes(x="", y=prop_ventes, fill=categ)) +
  geom_bar(stat="identity", width=1, color="white") +
  coord_polar("y", start=0) +
  theme_void() +
  ggtitle("Ventes par catégorie")

La catégorie la plus vendue est la catégorie 0 alors que la moins vendue est la catégorie 2. En ce qui concerne la répartition du chiffre d’affaires, elle est beaucoup plus équilibrée que la répartition des ventes (cette différence est dûe aux écarts de prix constatés entre les catégories).

Évolution du chiffre d’affaires mensuel d’une année sur l’autre

Créons un dataframe nous permettant d’obtenir les ventes et chiffre d’affaires quotidiens :

library(magrittr)
library(dplyr)

data_per_date <- data %>% group_by(date, annee, mois, jour) %>% summarise(ca = sum(price), nb_ventes = n())

## `summarise()` has grouped output by 'date', 'annee', 'mois'. You can override
## using the `.groups` argument.

data_per_date$annee <- as.factor(data_per_date$annee)
head(data_per_date)

## # A tibble: 6 × 6
## # Groups:   date, annee, mois [6]
##   date       annee  mois  jour     ca nb_ventes
##   <date>     <fct> <dbl> <int>  <dbl>     <int>
## 1 2021-03-01 2021      3     1 16577.       963
## 2 2021-03-02 2021      3     2 15498.       940
## 3 2021-03-03 2021      3     3 15199.       911
## 4 2021-03-04 2021      3     4 15196.       903
## 5 2021-03-05 2021      3     5 17471.       943
## 6 2021-03-06 2021      3     6 15785.       961

Comparons maintenant les chiffres d’affaires réalisés chaque mois sur les années N-1 et N :

library(magrittr)
library(dplyr)
library(ggplot2)
library(graphics)

par(mfrow=c(3,4))

data_per_date[(data_per_date$mois == "3"),] %>%
  ggplot( aes(x=jour, y=ca, group=annee, color=annee)) +
  xlab("Jour") +
  ylab("Chiffre d'affaires") +
  theme_ipsum() +
  ggtitle("Ventes des mois de mars 2021 et 2022") +
    geom_line()

data_per_date[(data_per_date$mois == "4"),] %>%
  ggplot( aes(x=jour, y=ca, group=annee, color=annee)) +
  xlab("Jour") +
  ylab("Chiffre d'affaires") +
  theme_ipsum() +
  ggtitle("Ventes des mois d'avril 2021 et 2022") +
    geom_line()

data_per_date[(data_per_date$mois == "5"),] %>%
  ggplot( aes(x=jour, y=ca, group=annee, color=annee)) +
  xlab("Jour") +
  ylab("Chiffre d'affaires") +
  theme_ipsum() +
  ggtitle("Ventes des mois de mai 2021 et 2022") +
    geom_line()

data_per_date[(data_per_date$mois == "6"),] %>%
  ggplot( aes(x=jour, y=ca, group=annee, color=annee)) +
  xlab("Jour") +
  ylab("Chiffre d'affaires") +
  theme_ipsum() +
  ggtitle("Ventes des mois de juin 2021 et 2022") +
    geom_line()

data_per_date[(data_per_date$mois == "7"),] %>%
  ggplot( aes(x=jour, y=ca, group=annee, color=annee)) +
  xlab("Jour") +
  ylab("Chiffre d'affaires") +
  theme_ipsum() +
  ggtitle("Ventes des mois de juillet 2021 et 2022") +
    geom_line()

data_per_date[(data_per_date$mois == "8"),] %>%
  ggplot( aes(x=jour, y=ca, group=annee, color=annee)) +
  xlab("Jour") +
  ylab("Chiffre d'affaires") +
  theme_ipsum() +
  ggtitle("Ventes des mois d'août 2021 et 2022") +
    geom_line()

data_per_date[(data_per_date$mois == "9"),] %>%
  ggplot( aes(x=jour, y=ca, group=annee, color=annee)) +
  xlab("Jour") +
  ylab("Chiffre d'affaires") +
  theme_ipsum() +
  ggtitle("Ventes des mois de septembre 2021 et 2022") +
    geom_line()

data_per_date[(data_per_date$mois == "10"),] %>%
  ggplot( aes(x=jour, y=ca, group=annee, color=annee)) +
  xlab("Jour") +
  ylab("Chiffre d'affaires") +
  theme_ipsum() +
  ggtitle("Ventes des mois d'octobre 2021 et 2022") +
    geom_line()

data_per_date[(data_per_date$mois == "11"),] %>%
  ggplot( aes(x=jour, y=ca, group=annee, color=annee)) +
  xlab("Jour") +
  ylab("Chiffre d'affaires") +
  theme_ipsum() +
  ggtitle("Ventes des mois de novembre 2021 et 2022") +
    geom_line()

data_per_date[(data_per_date$mois == "12"),] %>%
  ggplot( aes(x=jour, y=ca, group=annee, color=annee)) +
  xlab("Jour") +
  ylab("Chiffre d'affaires") +
  theme_ipsum() +
  ggtitle("Ventes des mois de décembre 2021 et 2022") +
    geom_line()

data_per_date[(data_per_date$mois == "1"),] %>%
  ggplot( aes(x=jour, y=ca, group=annee, color=annee)) +
  xlab("Jour") +
  ylab("Chiffre d'affaires") +
  theme_ipsum() +
  ggtitle("Ventes des mois de janvier 2022 et 2023") +
    geom_line()

data_per_date[(data_per_date$mois == "2"),] %>%
  ggplot( aes(x=jour, y=ca, group=annee, color=annee)) +
  xlab("Jour") +
  ylab("Chiffre d'affaires") +
  theme_ipsum() +
  ggtitle("Ventes des mois de février 2022 et 2023") +
    geom_line()

Faisons de même pour l’intégralité des années N-1 et N :

library(magrittr)
library(dplyr)
library(ggplot2)
library(graphics)

n_mois <- rep(c("01-Mar","02-Apr","03-May","04-Jun","05-Jul","06-Aug","07-Sep","08-Oct","09-Nov","10-Dec","11-Jan","12-Feb"),2)
data_per_month$n_mois <- n_mois
n_annee <- rep(c("Année 1", "Année 2"), each=12)
data_per_month$n_annee <- n_annee

data_per_month %>%
  ggplot(aes(x=n_mois, y=ca, group=n_annee, color=n_annee)) +
  xlab("Mois") +
  ylab("Chiffre d'affaires") +
  theme_ipsum() +
  ggtitle("Ventes des années 2021 et 2022 (années glissantes)") +
    geom_line()

Nous pouvons observer que le chiffre d’affaires est relativement stable au cours de l’année et ne semble pas connaitre de phénomène de saisonnalité.

Répartition du CA par client

Commençons par calculer le chiffre d’affaires par client :

library(magrittr)
library(dplyr)

ca_per_client <- unique(data %>% group_by(client_id) %>% summarise(age=2023-birth, ca=sum(price), panier_moyen = round(mean(price), digits = 2), nb_achats = n()))

## `summarise()` has grouped output by 'client_id'. You can override using the
## `.groups` argument.

ca_per_client <- arrange(ca_per_client, desc(ca_per_client$ca))
lc_ca_client <- as.vector(cumsum(sort(ca_per_client$ca))/sum(ca_per_client$ca))

library(ineq)

lorenz_curve <- Lc(lc_ca_client, n=rep(1,8600), plot=FALSE)
plot(lorenz_curve, xlab="Parts cumulées des achats", ylab="Parts cumulées du chiffre d'affaires", main="Répartition du chiffre d'affaires par client")

library(DescTools)

## Registered S3 methods overwritten by 'DescTools':
##   method   from
##   lines.Lc ineq
##   plot.Lc  ineq

## 
## Attaching package: 'DescTools'

## The following objects are masked from 'package:ineq':
## 
##     Atkinson, Gini, Herfindahl, Lc, Rosenbluth

Gini(lc_ca_client, n=rep(1,8600))

## [1] 0.5035666

Petit zoom sur les clients

head(ca_per_client, n=10L)

## # A tibble: 10 × 5
## # Groups:   client_id [10]
##    client_id   age      ca panier_moyen nb_achats
##    <chr>     <dbl>   <dbl>        <dbl>     <int>
##  1 c_1609       43 324033.         12.7     25488
##  2 c_4958       24 289760.         55.8      5195
##  3 c_6714       55 153669.         16.7      9187
##  4 c_3454       54 113673.         16.8      6773
##  5 c_3263       38   5277.         13.1       403
##  6 c_1570       44   5272.         14.3       369
##  7 c_2899       29   5214.         49.7       105
##  8 c_2140       46   5209.         13.0       402
##  9 c_7319       49   5156.         13.9       371
## 10 c_8026       45   5094.         13.5       377

À en juger par le montant du chiffre d’affaires des 4 plus gros acheteurs ainsi que leurs nombres de commandes, nous pouvons supposer qu’il s’agisse de revendeurs. Il serait judicieux de leur proposer un accompagnement personnalisé dans la mesure où leurs besoins seront différents des besoins de clients particuliers.

Petit zoom sur les produits

library(dplyr)
library(magrittr)

mean(produits[(produits$categ == "0"),]$price)

## [1] 11.72728

mean(produits[(produits$categ == "1"),]$price)

## [1] 25.53142

mean(produits[(produits$categ == "2"),]$price)

## [1] 108.3547

Comme nous l’avions déjà constaté en comparant la répartition du chiffre d’afffaires ainsi que celle des ventes entre les différentes catégories, les écarts de prix moyens entre celles-ci sont plûtot importants (notamment entre les catégories 1 et 2 et la catégorie 3).

library(dplyr)
library(magrittr)

data_per_prod <- data %>% group_by(id_prod) %>% summarise(nb_ventes=n())

Top 10 des meilleures ventes

library(dplyr)

top <- arrange(data_per_prod, desc(data_per_prod$nb_ventes))
head(top, n=10L)

## # A tibble: 10 × 2
##    id_prod nb_ventes
##    <chr>       <int>
##  1 1_369        2252
##  2 1_417        2189
##  3 1_414        2180
##  4 1_498        2128
##  5 1_425        2096
##  6 1_403        1960
##  7 1_412        1951
##  8 1_413        1945
##  9 1_406        1939
## 10 1_407        1935

Les 10 références les plus vendues appartiennent toutes à la catégorie 1.

Top 10 des plus mauvaises ventes

library(dplyr)

flop <- arrange(data_per_prod, data_per_prod$nb_ventes)
head(flop, n=10L)

## # A tibble: 10 × 2
##    id_prod nb_ventes
##    <chr>       <int>
##  1 0_1151          1
##  2 0_1284          1
##  3 0_1379          1
##  4 0_1498          1
##  5 0_1539          1
##  6 0_1601          1
##  7 0_1633          1
##  8 0_1683          1
##  9 0_1728          1
## 10 0_2201          1

Les 10 références les moins vendues, quant à elles, appartiennent toutes à la catégorie 0.

Evaluation de la tendance globale

Commençons par créer de nouveaux dataframes en ne sélectionnant que les colonnes qui nous intéressent :

library(dplyr)

dpm <- subset(data_per_month, select=c("annee_mois","ca","nb_ventes"))
head(dpm)

## # A tibble: 6 × 3
## # Groups:   annee_mois [6]
##   annee_mois      ca nb_ventes
##        <mth>   <dbl>     <int>
## 1   2021 Mar 482546.     28610
## 2   2021 Apr 476273.     28457
## 3   2021 May 493037.     28293
## 4   2021 Jun 484171.     26857
## 5   2021 Jul 482882.     24742
## 6   2021 Aug 482390.     25659

dpd <- subset(data_per_date, select=c("date", "ca", "nb_ventes"))
head(dpd)

## # A tibble: 6 × 3
## # Groups:   date [6]
##   date           ca nb_ventes
##   <date>      <dbl>     <int>
## 1 2021-03-01 16577.       963
## 2 2021-03-02 15498.       940
## 3 2021-03-03 15199.       911
## 4 2021-03-04 15196.       903
## 5 2021-03-05 17471.       943
## 6 2021-03-06 15785.       961

Décomposition en moyennes mobiles mensuelles

library(stats)

dpm_ts <- ts(dpm, frequency = 12)
decompose(dpm_ts)

## $x
##       annee_mois       ca nb_ventes
## Jan 1        614 482546.2     28610
## Feb 1        615 476273.5     28457
## Mar 1        616 493037.3     28293
## Apr 1        617 484170.7     26857
## May 1        618 482882.3     24742
## Jun 1        619 482390.3     25659
## Jul 1        620 507381.4     33326
## Aug 1        621 320880.8     21606
## Sep 1        622 516285.0     28321
## Oct 1        623 525999.4     32464
## Nov 1        624 525397.6     29348
## Dec 1        625 535700.5     29605
## Jan 2        626 515585.5     29707
## Feb 2        627 493163.1     27616
## Mar 2        628 517320.2     29991
## Apr 2        629 496098.2     28511
## May 2        630 510923.8     28682
## Jun 2        631 506561.1     28552
## Jul 2        632 494220.1     28315
## Aug 2        633 508035.0     28974
## Sep 2        634 496793.9     28574
## Oct 2        635 510289.9     28625
## Nov 2        636 517622.6     28945
## Dec 2        637 456761.9     25552
## 
## $seasonal
##           Jan         Feb         Mar         Apr         May         Jun
## 1   6856.2207  -3764.2193   2645.3169  -4383.3226   1000.6464    769.3049
## 2   6856.2207  -3764.2193   2645.3169  -4383.3226   1000.6464    769.3049
## 3   6856.2207  -3764.2193   2645.3169  -4383.3226   1000.6464    769.3049
## 4   6856.2207  -3764.2193   2645.3169  -4383.3226   1000.6464    769.3049
## 5   6856.2207  -3764.2193   2645.3169  -4383.3226   1000.6464    769.3049
## 6   6856.2207  -3764.2193   2645.3169  -4383.3226   1000.6464    769.3049
##           Jul         Aug         Sep         Oct         Nov         Dec
## 1   8734.4704 -58036.0692   8753.2438  12822.8872  10950.8182  13650.7026
## 2   8734.4704 -58036.0692   8753.2438  12822.8872  10950.8182  13650.7026
## 3   8734.4704 -58036.0692   8753.2438  12822.8872  10950.8182  13650.7026
## 4   8734.4704 -58036.0692   8753.2438  12822.8872  10950.8182  13650.7026
## 5   8734.4704 -58036.0692   8753.2438  12822.8872  10950.8182  13650.7026
## 6   8734.4704 -58036.0692   8753.2438  12822.8872  10950.8182  13650.7026
## 
## $trend
##       [,1]     [,2]     [,3]
## Jan 1   NA       NA       NA
## Feb 1   NA       NA       NA
## Mar 1   NA       NA       NA
## Apr 1   NA       NA       NA
## May 1   NA       NA       NA
## Jun 1   NA       NA       NA
## Jul 1  620 487455.4 28153.04
## Aug 1  621 489535.8 28163.71
## Sep 1  622 491251.3 28199.42
## Oct 1  623 492760.1 28339.08
## Nov 1  624 494425.4 28572.17
## Dec 1  625 496600.9 28856.87
## Jan 2  626 497059.7 28768.62
## Feb 2  627 504309.4 28866.83
## Mar 2  628 511295.3 29184.37
## Apr 2  629 509828.7 29034.96
## May 2  630 508850.1 28858.21
## Jun 2  631 505237.1 28672.54
## Jul 2   NA       NA       NA
## Aug 2   NA       NA       NA
## Sep 2   NA       NA       NA
## Oct 2   NA       NA       NA
## Nov 2   NA       NA       NA
## Dec 2   NA       NA       NA
## 
## $random
##       x - seasonal.x.annee_mois x - seasonal.x.ca x - seasonal.x.nb_ventes
## Jan 1                        NA                NA                       NA
## Feb 1                        NA                NA                       NA
## Mar 1                        NA                NA                       NA
## Apr 1                        NA                NA                       NA
## May 1                        NA                NA                       NA
## Jun 1                        NA                NA                       NA
## Jul 1                -8734.4704        11191.5497               -3561.5120
## Aug 1                58036.0692      -110618.8627               51478.3609
## Sep 1                -8753.2438        16280.4716               -8631.6605
## Oct 1               -12822.8872        20416.4252               -8697.9706
## Nov 1               -10950.8182        20021.3705              -10174.9849
## Dec 1               -13650.7026        25448.8477              -12902.5776
## Jan 2                -6856.2207        11669.6338               -5917.8457
## Feb 2                 3764.2193        -7382.0378                2513.3859
## Mar 2                -2645.3169         3379.5761               -1838.6919
## Apr 2                 4383.3226        -9347.1195                3859.3643
## May 2                -1000.6464         1073.0684               -1176.8547
## Jun 2                 -769.3049          554.7189                -889.8466
## Jul 2                        NA                NA                       NA
## Aug 2                        NA                NA                       NA
## Sep 2                        NA                NA                       NA
## Oct 2                        NA                NA                       NA
## Nov 2                        NA                NA                       NA
## Dec 2                        NA                NA                       NA
## 
## $figure
##  [1]   6856.2207  -3764.2193   2645.3169  -4383.3226   1000.6464    769.3049
##  [7]   8734.4704 -58036.0692   8753.2438  12822.8872  10950.8182  13650.7026
## 
## $type
## [1] "additive"
## 
## attr(,"class")
## [1] "decomposed.ts"

plot(dpm_ts)

library(zoo)

rollmean(dpm_ts, 3)

##       annee_mois       ca nb_ventes
## Feb 1        615 483952.3  28453.33
## Mar 1        616 484493.8  27869.00
## Apr 1        617 486696.7  26630.67
## May 1        618 483147.8  25752.67
## Jun 1        619 490884.7  27909.00
## Jul 1        620 436884.2  26863.67
## Aug 1        621 448182.4  27751.00
## Sep 1        622 454388.4  27463.67
## Oct 1        623 522560.7  30044.33
## Nov 1        624 529032.5  30472.33
## Dec 1        625 525561.2  29553.33
## Jan 2        626 514816.4  28976.00
## Feb 2        627 508689.6  29104.67
## Mar 2        628 502193.9  28706.00
## Apr 2        629 508114.1  29061.33
## May 2        630 504527.7  28581.67
## Jun 2        631 503901.7  28516.33
## Jul 2        632 502938.7  28613.67
## Aug 2        633 499683.0  28621.00
## Sep 2        634 505039.6  28724.33
## Oct 2        635 508235.5  28714.67
## Nov 2        636 494891.5  27707.33

plot(rollmean(dpm_ts, 3))

Décomposition en moyennes mobiles quotidiennes

library(stats)

dpd_ts <- ts(dpd, frequency = 365)
decompose(dpd_ts)

## $x
## Time Series:
## Start = c(1, 1) 
## End = c(2, 365) 
## Frequency = 365 
##           date        ca nb_ventes
## 1.000000 18687 16576.947       963
## 1.002740 18688 15498.177       940
## 1.005479 18689 15198.690       911
## 1.008219 18690 15196.070       903
## 1.010959 18691 17471.370       943
## 1.013699 18692 15785.280       961
## 1.016438 18693 14771.927       906
## 1.019178 18694 15679.530       928
## 1.021918 18695 15710.510       952
## 1.024658 18696 15496.870       942
## 1.027397 18697 14801.140       914
## 1.030137 18698 14448.580       888
## 1.032877 18699 14324.840       884
## 1.035616 18700 15231.770       914
## 1.038356 18701 16141.660       918
## 1.041096 18702 15971.740       945
## 1.043836 18703 16025.470       958
## 1.046575 18704 15060.790       922
## 1.049315 18705 15545.770       917
## 1.052055 18706 15853.990       945
## 1.054795 18707 17459.660       985
## 1.057534 18708 15657.050       919
## 1.060274 18709 16040.277       921
## 1.063014 18710 15101.700       880
## 1.065753 18711 15162.180       927
## 1.068493 18712 16176.280       919
## 1.071233 18713 15238.090       889
## 1.073973 18714 16100.675       916
## 1.076712 18715 14460.495       883
## 1.079452 18716 14827.237       898
## 1.082192 18717 15531.390       919
## 1.084932 18718 15520.780       916
## 1.087671 18719 15982.777       961
## 1.090411 18720 15516.037       948
## 1.093151 18721 16115.580       886
## 1.095890 18722 17472.730       983
## 1.098630 18723 16811.747       960
## 1.101370 18724 15507.217       932
## 1.104110 18725 15002.420       918
## 1.106849 18726 15749.197       925
## 1.109589 18727 15260.829       937
## 1.112329 18728 16429.510       989
## 1.115068 18729 15916.890       968
## 1.117808 18730 15594.560       916
## 1.120548 18731 15956.470       936
## 1.123288 18732 15552.730       954
## 1.126027 18733 14773.480       916
## 1.128767 18734 15836.347       941
## 1.131507 18735 14954.187       946
## 1.134247 18736 15513.760       928
## 1.136986 18737 15597.650       934
## 1.139726 18738 15404.050       922
## 1.142466 18739 17043.357      1003
## 1.145205 18740 15242.040       900
## 1.147945 18741 16572.720       968
## 1.150685 18742 17448.827      1014
## 1.153425 18743 15306.200       936
## 1.156164 18744 16008.117       964
## 1.158904 18745 15243.840       978
## 1.161644 18746 16838.330       993
## 1.164384 18747 16101.100       985
## 1.167123 18748 15763.577       909
## 1.169863 18749 15438.080       892
## 1.172603 18750 15432.460       944
## 1.175342 18751 16749.740       920
## 1.178082 18752 15709.100       906
## 1.180822 18753 14247.347       890
## 1.183562 18754 18366.000       943
## 1.186301 18755 18696.030      1002
## 1.189041 18756 16214.920       933
## 1.191781 18757 15992.680       897
## 1.194521 18758 16380.577       927
## 1.197260 18759 16229.017       946
## 1.200000 18760 15287.080       911
## 1.202740 18761 14787.810       886
## 1.205479 18762 15149.990       890
## 1.208219 18763 15365.740       879
## 1.210959 18764 15732.850       882
## 1.213699 18765 15373.610       875
## 1.216438 18766 15575.890       915
## 1.219178 18767 15562.377       886
## 1.221918 18768 15281.247       888
## 1.224658 18769 15727.717       897
## 1.227397 18770 15598.260       910
## 1.230137 18771 15356.870       896
## 1.232877 18772 15614.140       881
## 1.235616 18773 15803.440       901
## 1.238356 18774 16225.990       908
## 1.241096 18775 17106.930       966
## 1.243836 18776 15770.210       957
## 1.246575 18777 16638.340       948
## 1.249315 18778 15859.267       908
## 1.252055 18779 16399.150       921
## 1.254795 18780 14904.190       897
## 1.257534 18781 16220.190       914
## 1.260274 18782 14586.310       855
## 1.263014 18783 16585.227       922
## 1.265753 18784 17425.790       916
## 1.268493 18785 16390.300       905
## 1.271233 18786 16391.760       880
## 1.273973 18787 15463.277       894
## 1.276712 18788 15395.510       901
## 1.279452 18789 15999.890       889
## 1.282192 18790 16422.500       896
## 1.284932 18791 15097.620       855
## 1.287671 18792 16540.950       916
## 1.290411 18793 15298.230       867
## 1.293151 18794 15806.237       890
## 1.295890 18795 15916.985       880
## 1.298630 18796 16731.340       876
## 1.301370 18797 15957.820       881
## 1.304110 18798 16151.340       896
## 1.306849 18799 15621.527       867
## 1.309589 18800 16211.960       943
## 1.312329 18801 17367.440       911
## 1.315068 18802 18110.887       930
## 1.317808 18803 16468.730       897
## 1.320548 18804 16304.720       890
## 1.323288 18805 16241.650       892
## 1.326027 18806 15487.720       920
## 1.328767 18807 16471.110       881
## 1.331507 18808 16200.290       875
## 1.334247 18809 16569.450       876
## 1.336986 18810 17572.170       844
## 1.339726 18811 15532.297       840
## 1.342466 18812 15887.420       824
## 1.345205 18813 15172.060       796
## 1.347945 18814 15246.880       798
## 1.350685 18815 16626.110       837
## 1.353425 18816 16783.010       836
## 1.356164 18817 15009.220       785
## 1.358904 18818 14424.357       766
## 1.361644 18819 14065.270       815
## 1.364384 18820 17112.807       810
## 1.367123 18821 15991.180       806
## 1.369863 18822 15300.240       793
## 1.372603 18823 15460.757       789
## 1.375342 18824 16061.330       852
## 1.378082 18825 16379.640       811
## 1.380822 18826 15443.330       813
## 1.383562 18827 15505.870       774
## 1.386301 18828 15638.590       784
## 1.389041 18829 14602.710       762
## 1.391781 18830 14341.260       762
## 1.394521 18831 14938.600       782
## 1.397260 18832 13792.210       725
## 1.400000 18833 15411.280       749
## 1.402740 18834 13846.530       755
## 1.405479 18835 16050.540       801
## 1.408219 18836 15848.940       780
## 1.410959 18837 16919.190       814
## 1.413699 18838 15955.340       784
## 1.416438 18839 15393.720       779
## 1.419178 18840 15149.710       815
## 1.421918 18841 15408.890       804
## 1.424658 18842 15673.850       831
## 1.427397 18843 15153.990       799
## 1.430137 18844 15129.350       810
## 1.432877 18845 16177.935       831
## 1.435616 18846 14828.950       804
## 1.438356 18847 15546.810       805
## 1.441096 18848 17261.760       872
## 1.443836 18849 15136.570       834
## 1.446575 18850 14454.590       799
## 1.449315 18851 14945.570       794
## 1.452055 18852 14622.440       810
## 1.454795 18853 15709.457       845
## 1.457534 18854 15921.357       802
## 1.460274 18855 14818.927       853
## 1.463014 18856 16882.070       855
## 1.465753 18857 14957.600       827
## 1.468493 18858 15670.597       847
## 1.471233 18859 15066.800       823
## 1.473973 18860 14967.760       847
## 1.476712 18861 15632.607       824
## 1.479452 18862 15344.147       835
## 1.482192 18863 15416.770       831
## 1.484932 18864 15649.737       804
## 1.487671 18865 16020.950       848
## 1.490411 18866 17073.440       892
## 1.493151 18867 15871.060       808
## 1.495890 18868 14811.700       805
## 1.498630 18869 16746.040       883
## 1.501370 18870 16338.900       822
## 1.504110 18871 15568.715       963
## 1.506849 18872 16015.980       973
## 1.509589 18873 16132.607      1011
## 1.512329 18874 14771.230       933
## 1.515068 18875 15240.280       959
## 1.517808 18876 15961.617       983
## 1.520548 18877 16763.027      1003
## 1.523288 18878 15474.500       980
## 1.526027 18879 16527.850      1035
## 1.528767 18880 16856.427      1050
## 1.531507 18881 17273.777      1101
## 1.534247 18882 16992.060      1101
## 1.536986 18883 16146.610      1084
## 1.539726 18884 17256.780      1115
## 1.542466 18885 17926.540      1144
## 1.545205 18886 16711.950      1111
## 1.547945 18887 16316.177      1120
## 1.550685 18888 16712.770      1091
## 1.553425 18889 17109.807      1147
## 1.556164 18890 16725.860      1128
## 1.558904 18891 16238.970      1127
## 1.561644 18892 16739.790      1162
## 1.564384 18893 18149.487      1242
## 1.567123 18894 18027.800      1219
## 1.569863 18895 19092.017      1298
## 1.572603 18896 17431.017      1188
## 1.575342 18897 18542.390      1242
## 1.578082 18898 18136.650      1237
## 1.580822 18899 18410.710      1268
## 1.583562 18900 18128.010      1311
## 1.586301 18901 17058.340      1045
## 1.589041 18902  9182.130       689
## 1.591781 18903  8993.050       679
## 1.594521 18904  9303.135       641
## 1.597260 18905  9390.460       632
## 1.600000 18906  9490.690       728
## 1.602740 18907  8191.080       623
## 1.605479 18908 10207.350       713
## 1.608219 18909  9425.360       675
## 1.610959 18910  8825.650       629
## 1.613699 18911 10230.560       684
## 1.616438 18912  8971.150       663
## 1.619178 18913  9427.220       669
## 1.621918 18914  9469.780       646
## 1.624658 18915  9410.557       672
## 1.627397 18916 10229.020       706
## 1.630137 18917  9557.980       669
## 1.632877 18918  9083.720       645
## 1.635616 18919  8625.165       604
## 1.638356 18920  9032.917       597
## 1.641096 18921  9571.790       653
## 1.643836 18922  8467.860       607
## 1.646575 18923  8910.640       602
## 1.649315 18924  9923.190       628
## 1.652055 18925  8076.040       577
## 1.654795 18926  9032.517       628
## 1.657534 18927  9789.530       579
## 1.660274 18928 14758.420       894
## 1.663014 18929 16453.630       971
## 1.665753 18930 15386.780       920
## 1.668493 18931 16405.120       938
## 1.671233 18932 17821.737       936
## 1.673973 18933 17866.447       977
## 1.676712 18934 15875.140       907
## 1.679452 18935 16512.380       914
## 1.682192 18936 16919.800       961
## 1.684932 18937 14841.960       907
## 1.687671 18938 16986.060       951
## 1.690411 18939 17175.840       970
## 1.693151 18940 15760.690       906
## 1.695890 18941 17782.250       926
## 1.698630 18942 17048.857       969
## 1.701370 18943 17064.787       957
## 1.704110 18944 17606.820       989
## 1.706849 18945 17149.230       979
## 1.709589 18946 16761.280       948
## 1.712329 18947 16628.410       973
## 1.715068 18948 17598.960       958
## 1.717808 18949 17984.410       954
## 1.720548 18950 16142.767       906
## 1.723288 18951 16551.527       896
## 1.726027 18952 17168.677       925
## 1.728767 18953 17380.420       921
## 1.731507 18954 18967.820       985
## 1.734247 18955 17154.997       926
## 1.736986 18956 17842.170       917
## 1.739726 18957 17963.110       967
## 1.742466 18958 16772.657       928
## 1.745205 18959 18105.150       961
## 1.747945 18960 17314.080       945
## 1.750685 18961 19536.567       962
## 1.753425 18962 16633.450       982
## 1.756164 18963 17100.360       978
## 1.758904 18964 16381.327       976
## 1.761644 18965 17451.250       969
## 1.764384 18966 17041.040      1012
## 1.767123 18967 16877.810      1005
## 1.769863 18968 17022.010      1012
## 1.772603 18969 17245.770      1041
## 1.775342 18970 16383.200      1047
## 1.778082 18971 17177.305       994
## 1.780822 18972 16487.280      1017
## 1.783562 18973 16273.500      1010
## 1.786301 18974 16950.160      1030
## 1.789041 18975 16696.547      1019
## 1.791781 18976 16082.920      1015
## 1.794521 18977 16880.967      1060
## 1.797260 18978 17617.380      1060
## 1.800000 18979 16599.320      1049
## 1.802740 18980 16490.640      1027
## 1.805479 18981 16686.110      1049
## 1.808219 18982 15970.150      1035
## 1.810959 18983 16994.520      1075
## 1.813699 18984 17266.747      1085
## 1.816438 18985 17547.470      1116
## 1.819178 18986 17127.980      1094
## 1.821918 18987 17094.070      1086
## 1.824658 18988 17524.300      1105
## 1.827397 18989 17503.800      1115
## 1.830137 18990 17363.147      1126
## 1.832877 18991 16681.530      1069
## 1.835616 18992 18847.310      1206
## 1.838356 18993 17106.610      1011
## 1.841096 18994 16345.090       951
## 1.843836 18995 17435.810       995
## 1.846575 18996 17200.827      1015
## 1.849315 18997 15962.430       932
## 1.852055 18998 17131.430       983
## 1.854795 18999 16366.520       963
## 1.857534 19000 17718.940      1018
## 1.860274 19001 15158.737       903
## 1.863014 19002 17299.840      1006
## 1.865753 19003 16587.820       943
## 1.868493 19004 17413.870       949
## 1.871233 19005 17451.420       957
## 1.873973 19006 17557.460       960
## 1.876712 19007 16977.657       943
## 1.879452 19008 16539.600       977
## 1.882192 19009 18203.860       975
## 1.884932 19010 16537.290       935
## 1.887671 19011 16309.910       880
## 1.890411 19012 17032.000       930
## 1.893151 19013 17782.997       945
## 1.895890 19014 15694.600       889
## 1.898630 19015 16673.880       941
## 1.901370 19016 17338.520       933
## 1.904110 19017 15761.250       905
## 1.906849 19018 16822.907       961
## 1.909589 19019 16040.470       881
## 1.912329 19020 18039.630       925
## 1.915068 19021 18505.530       949
## 1.917808 19022 17755.050       944
## 1.920548 19023 16645.670       849
## 1.923288 19024 17152.320       981
## 1.926027 19025 19182.240      1074
## 1.928767 19026 18144.550      1002
## 1.931507 19027 19425.140      1029
## 1.934247 19028 18844.620      1002
## 1.936986 19029 19267.360      1069
## 1.939726 19030 18922.397      1079
## 1.942466 19031 19097.887      1027
## 1.945205 19032 18582.980      1037
## 1.947945 19033 19894.900      1101
## 1.950685 19034 19925.417      1062
## 1.953425 19035 19632.280      1080
## 1.956164 19036 19135.350      1081
## 1.958904 19037 19997.230      1106
## 1.961644 19038 20314.617      1087
## 1.964384 19039 19760.117      1101
## 1.967123 19040 19088.340      1044
## 1.969863 19041 18986.490      1045
## 1.972603 19042 18516.250      1037
## 1.975342 19043 19799.065      1107
## 1.978082 19044 18463.570      1063
## 1.980822 19045 18048.720      1030
## 1.983562 19046 19013.450      1031
## 1.986301 19047 20350.427      1078
## 1.989041 19048 18205.415       990
## 1.991781 19049 19908.560      1103
## 1.994521 19050 19170.810      1110
## 1.997260 19051 18869.997      1049
## 2.000000 19052 17152.320       981
## 2.002740 19053 19331.220      1076
## 2.005479 19054 17286.830       993
## 2.008219 19055 14735.290       857
## 2.010959 19056 15962.430       932
## 2.013699 19057 16662.767       958
## 2.016438 19058 18922.397      1079
## 2.019178 19059 16093.800       876
## 2.021918 19060 16207.630       912
## 2.024658 19061 17931.230       928
## 2.027397 19062 16082.617       923
## 2.030137 19063 14796.590       792
## 2.032877 19064 14473.820       886
## 2.035616 19065 16696.547      1019
## 2.038356 19066 15552.730       954
## 2.041096 19067 15365.740       879
## 2.043836 19068 15768.005       878
## 2.046575 19069 19433.430      1051
## 2.049315 19070 15356.890       772
## 2.052055 19071 16002.970       947
## 2.054795 19072 14453.730       760
## 2.057534 19073 16590.810      1160
## 2.060274 19074 18149.487      1242
## 2.063014 19075 17338.520       933
## 2.065753 19076 18943.037      1296
## 2.068493 19077 16020.950       848
## 2.071233 19078 16077.010       906
## 2.073973 19079 19018.977      1051
## 2.076712 19080 17363.147      1126
## 2.079452 19081 15955.340       784
## 2.082192 19082 15859.267       908
## 2.084932 19083 15298.690       817
## 2.087671 19084 17423.190       842
## 2.090411 19085 16132.607      1011
## 2.093151 19086 14920.210       935
## 2.095890 19087 16585.227       922
## 2.098630 19088 15879.975       827
## 2.101370 19089 16986.060       951
## 2.104110 19090 15325.520       978
## 2.106849 19091 15009.220       785
## 2.109589 19092 15856.749       945
## 2.112329 19093 16587.820       943
## 2.115068 19094 16229.017       946
## 2.117808 19095 15585.040       915
## 2.120548 19096 17149.230       979
## 2.123288 19097 15772.377       800
## 2.126027 19098 16104.197       894
## 2.128767 19099 17449.980       956
## 2.131507 19100 19284.450      1049
## 2.134247 19101 15670.597       847
## 2.136986 19102 16686.110      1049
## 2.139726 19103 15116.740       849
## 2.142466 19104 15632.607       824
## 2.145205 19105 18967.820       985
## 2.147945 19106 17729.840      1215
## 2.150685 19107 17299.847      1012
## 2.153425 19108 13995.510       757
## 2.156164 19109 16225.990       908
## 2.158904 19110 15997.920       782
## 2.161644 19111 18505.530       949
## 2.164384 19112 17755.050       944
## 2.167123 19113 15614.597       907
## 2.169863 19114 17717.467       975
## 2.172603 19115 16430.567      1015
## 2.175342 19116 17200.827      1015
## 2.178082 19117 16883.187       926
## 2.180822 19118 14396.327       892
## 2.183562 19119 16837.080       949
## 2.186301 19120 15176.540       976
## 2.189041 19121 16363.900       935
## 2.191781 19122 17326.285       996
## 2.194521 19123 17422.757      1103
## 2.197260 19124 16080.037       944
## 2.200000 19125 14771.780       890
## 2.202740 19126 15380.750       916
## 2.205479 19127 20463.597      1089
## 2.208219 19128 19760.117      1101
## 2.210959 19129 18352.840       977
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## 2.216438 19131 15993.787       904
## 2.219178 19132 15638.590       784
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## 2.295890 19160 16065.965       882
## 2.298630 19161 14808.620       825
## 2.301370 19162 16106.800       883
## 2.304110 19163 16002.360       894
## 2.306849 19164 18612.550      1065
## 2.309589 19165 17143.500      1077
## 2.312329 19166 15891.297       919
## 2.315068 19167 15399.660       884
## 2.317808 19168 17276.960      1096
## 2.320548 19169 16304.720       890
## 2.323288 19170 15901.560       799
## 2.326027 19171 17652.780      1117
## 2.328767 19172 16919.190       814
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## 2.334247 19174 17450.280       985
## 2.336986 19175 17249.340       980
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## 2.342466 19177 19276.160      1027
## 2.345205 19178 15709.100       906
## 2.347945 19179 19416.340      1071
## 2.350685 19180 18068.040       939
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## 2.380822 19191 15443.330       813
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## 2.386301 19193 15895.610       938
## 2.389041 19194 17019.697       923
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## 2.394521 19196 15195.167       833
## 2.397260 19197 20499.407      1080
## 2.400000 19198 17448.827      1014
## 2.402740 19199 13846.530       755
## 2.405479 19200 17226.340      1101
## 2.408219 19201 15848.940       780
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## 2.416438 19204 16008.247       910
## 2.419178 19205 15568.715       963
## 2.421918 19206 15867.000       971
## 2.424658 19207 14900.730       907
## 2.427397 19208 17051.847      1013
## 2.430137 19209 17621.710       985
## 2.432877 19210 15934.260       963
## 2.435616 19211 17170.990      1014
## 2.438356 19212 17569.960      1016
## 2.441096 19213 15760.690       906
## 2.443836 19214 17782.250       926
## 2.446575 19215 16899.877       967
## 2.449315 19216 16571.480       898
## 2.452055 19217 14948.640       853
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## 2.460274 19220 15955.217       892
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## 2.465753 19222 14954.187       946
## 2.468493 19223 15426.910       913
## 2.471233 19224 15787.570       786
## 2.473973 19225 18761.530      1067
## 2.476712 19226 14341.260       762
## 2.479452 19227 15493.127       837
## 2.482192 19228 17154.997       926
## 2.484932 19229 15411.280       749
## 2.487671 19230 17963.110       967
## 2.490411 19231 17073.440       892
## 2.493151 19232 17890.650       923
## 2.495890 19233 15919.190       959
## 2.498630 19234 16532.550      1067
## 2.501370 19235 16347.710       845
## 2.504110 19236 16718.430       878
## 2.506849 19237 16494.070       953
## 2.509589 19238 16220.190       914
## 2.512329 19239 15047.090       901
## 2.515068 19240 17190.020      1014
## 2.517808 19241 15785.280       961
## 2.520548 19242 16366.520       963
## 2.523288 19243 15002.420       918
## 2.526027 19244 17261.760       872
## 2.528767 19245 17177.305       994
## 2.531507 19246 14603.570       801
## 2.534247 19247 15243.530       798
## 2.536986 19248 18986.370      1079
## 2.539726 19249 16689.930       918
## 2.542466 19250 20314.617      1087
## 2.545205 19251 14967.907       855
## 2.547945 19252 16065.965       882
## 2.550685 19253 16388.310       933
## 2.553425 19254 15505.870       774
## 2.556164 19255 15746.630       936
## 2.558904 19256 16387.950      1129
## 2.561644 19257 17143.500      1077
## 2.564384 19258 18447.447      1246
## 2.567123 19259 15416.770       831
## 2.569863 19260 17299.847      1012
## 2.572603 19261 15803.440       901
## 2.575342 19262 16772.657       928
## 2.578082 19263 15722.080       806
## 2.580822 19264 16918.050       887
## 2.583562 19265 16532.550      1067
## 2.586301 19266 16576.947       963
## 2.589041 19267 16494.070       953
## 2.591781 19268 15962.977       954
## 2.594521 19269 14586.310       855
## 2.597260 19270 15129.350       810
## 2.600000 19271 16110.597       985
## 2.602740 19272 18217.020       941
## 2.605479 19273 15151.400       920
## 2.608219 19274 15463.277       894
## 2.610959 19275 15843.700       895
## 2.613699 19276 16082.617       923
## 2.616438 19277 15916.890       968
## 2.619178 19278 17600.400       959
## 2.621918 19279 17408.480       958
## 2.624658 19280 15460.757       789
## 2.627397 19281 15365.740       879
## 2.630137 19282 17449.980       956
## 2.632877 19283 16748.300      1051
## 2.635616 19284 17258.787      1149
## 2.638356 19285 19352.125      1101
## 2.641096 19286 14602.710       762
## 2.643836 19287 16590.810      1160
## 2.646575 19288 16822.860       943
## 2.649315 19289 15101.700       880
## 2.652055 19290 17448.827      1014
## 2.654795 19291 16020.950       848
## 2.657534 19292 17371.400       896
## 2.660274 19293 18039.630       925
## 2.663014 19294 14960.680       807
## 2.665753 19295 16399.060       989
## 2.668493 19296 16496.690       847
## 2.671233 19297 17106.610      1011
## 2.673973 19298 17423.190       842
## 2.676712 19299 16232.347       974
## 2.679452 19300 15153.990       799
## 2.682192 19301 18546.660       998
## 2.684932 19302 16513.787       956
## 2.687671 19303 19071.377      1081
## 2.690411 19304 17245.770      1041
## 2.693151 19305 15605.677       909
## 2.695890 19306 14424.357       766
## 2.698630 19307 14363.230       819
## 2.701370 19308 16992.060      1101
## 2.704110 19309 18688.410      1075
## 2.706849 19310 15709.457       845
## 2.709589 19311 15992.680       916
## 2.712329 19312 16120.720       947
## 2.715068 19313 17031.050       857
## 2.717808 19314 14954.187       946
## 2.720548 19315 15247.810       913
## 2.723288 19316 16402.547       894
## 2.726027 19317 15430.227       890
## 2.728767 19318 15843.580       891
## 2.731507 19319 17367.440       911
## 2.734247 19320 16870.680       972
## 2.736986 19321 15761.250       905
## 2.739726 19322 17282.037      1186
## 2.742466 19323 15794.710       886
## 2.745205 19324 18721.017      1047
## 2.747945 19325 16769.070       885
## 2.750685 19326 18128.010      1311
## 2.753425 19327 16427.967       961
## 2.756164 19328 16345.090       951
## 2.758904 19329 15532.297       840
## 2.761644 19330 16600.760       918
## 2.764384 19331 16883.187       926
## 2.767123 19332 17425.790       916
## 2.769863 19333 17170.990      1014
## 2.772603 19334 19544.827      1033
## 2.775342 19335 16383.200      1047
## 2.778082 19336 18080.210       930
## 2.780822 19337 14065.270       815
## 2.783562 19338 16065.870       970
## 2.786301 19339 17606.820       989
## 2.789041 19340 20146.210      1108
## 2.791781 19341 16977.657       943
## 2.794521 19342 15220.420       922
## 2.797260 19343 19386.300      1048
## 2.800000 19344 16712.770      1091
## 2.802740 19345 15356.890       772
## 2.805479 19346 15853.990       945
## 2.808219 19347 14453.730       760
## 2.810959 19348 15578.737       895
## 2.813699 19349 14938.600       782
## 2.816438 19350 15416.770       831
## 2.819178 19351 16170.770       893
## 2.821918 19352 13995.510       757
## 2.824658 19353 19319.790      1112
## 2.827397 19354 15636.700       922
## 2.830137 19355 14960.680       807
## 2.832877 19356 16638.340       948
## 2.835616 19357 15393.720       779
## 2.838356 19358 16576.947       963
## 2.841096 19359 17572.170       844
## 2.843836 19360 17286.830       993
## 2.846575 19361 17196.680       926
## 2.849315 19362 16472.860       955
## 2.852055 19363 15139.920       911
## 2.854795 19364 15126.910       808
## 2.857534 19365 19395.847      1031
## 2.860274 19366 18434.000      1035
## 2.863014 19367 15843.700       895
## 2.865753 19368 16736.800       945
## 2.868493 19369 17264.890       947
## 2.871233 19370 15991.180       806
## 2.873973 19371 17256.780      1115
## 2.876712 19372 16612.300       946
## 2.879452 19373 14773.480       916
## 2.882192 19374 15916.985       880
## 2.884932 19375 17686.450       950
## 2.887671 19376 15670.597       847
## 2.890411 19377 14917.820       821
## 2.893151 19378 17019.697       923
## 2.895890 19379 16360.940       945
## 2.898630 19380 16673.880       941
## 2.901370 19381 18027.800      1219
## 2.904110 19382 17842.170       917
## 2.906849 19383 17282.037      1186
## 2.909589 19384 17073.440       892
## 2.912329 19385 15573.100       804
## 2.915068 19386 16919.190       814
## 2.917808 19387 16638.340       948
## 2.920548 19388 16338.900       822
## 2.923288 19389 16718.430       878
## 2.926027 19390 17423.190       842
## 2.928767 19391 15681.277       842
## 2.931507 19392 14586.310       855
## 2.934247 19393 16111.410       934
## 2.936986 19394 14247.347       890
## 2.939726 19395 18217.020       941
## 2.942466 19396 15397.830       803
## 2.945205 19397 16065.940       931
## 2.947945 19398 14424.357       766
## 2.950685 19399 16636.260      1019
## 2.953425 19400 17562.850       951
## 2.956164 19401 14948.640       853
## 2.958904 19402 16689.930       918
## 2.961644 19403 15701.710       956
## 2.964384 19404 15806.237       890
## 2.967123 19405 15768.005       878
## 2.969863 19406 19135.470      1047
## 2.972603 19407 16106.800       883
## 2.975342 19408 16002.970       947
## 2.978082 19409 17931.977       947
## 2.980822 19410 15806.030       921
## 2.983562 19411 15242.040       900
## 2.986301 19412 15207.890       894
## 2.989041 19413 15761.250       905
## 2.991781 19414 16304.720       890
## 2.994521 19415 19170.810      1110
## 2.997260 19416 18105.150       961
## 
## $seasonal
## Time Series:
## Start = c(1, 1) 
## End = c(6, 365) 
## Frequency = 365 
##    [1]   286.9863748  1044.9239912   334.8216483  -561.4433654  -127.3880984
##    [6]   114.5016251   906.2141457  -104.1371511   -53.7951810   526.5592299
##   [11]   -91.8330557  -564.4173795  -637.9615281   149.1544421  -256.3734733
##   [16]  -343.0058934  -211.3798776  1069.9217920  -381.4790183  -107.3450822
##   [21]  -684.2863267   162.4766230   709.2311685   335.6176824   991.1810132
##   [26]  -129.4425733   -89.5247012   941.2128646   416.1724836  -164.4979158
##   [31]  -153.4777174  -368.9902719   348.0242420   -32.7553313  -468.8374633
##   [36]    76.8147119  -195.3397472   208.4084520  -345.5514019  -520.0222238
##   [41]  -189.8932632    46.4782627   -77.6852533  -309.2993604   225.7017081
##   [46]  -300.4507510  -164.1428971   300.2689501   935.2895437  -344.0317556
##   [51]    53.4235138  -546.2506937  -387.3222578   770.1587401   426.6282195
##   [56]   210.6712764  -984.7331935  -197.4895704  -322.7264837   565.7853337
##   [61]   315.4718634  -411.3331802   312.2896326  -102.7591096   154.5223125
##   [66]    18.5882420  -820.3528631    10.7118700  -535.3729951  -155.2761159
##   [71]   185.7233677   253.6864321  -243.6742139  -695.1703600  -483.5787826
##   [76]  1267.3040186  1038.2479206   528.5533021   -29.9759764  -281.0326730
##   [81]  -436.6571338    70.8562129  -191.5849993   773.1979231  -390.2778394
##   [86]  -356.6187253  -541.3677506    -2.9773147   555.1089094   361.8594615
##   [91]  -156.7906066  -561.3514193  -529.0671205  -684.0202869  -311.3767917
##   [96]  -282.2763143   666.8936309   981.3429941   163.1471676  -242.0540470
##  [101]  -393.0072433    68.0627567    33.9008729  -837.2494376   688.9133752
##  [106]  1185.2844619  -153.0505268  -538.8485518  -252.4711894  -692.1905360
##  [111]  -240.2724995  -270.1509013   657.7673362   173.7200028  -294.8902346
##  [116]  -468.0331205   230.6063316  -161.0826090  -322.6724173   365.4214732
##  [121]    21.7728430  -645.0593720   258.4381465   193.3353976  -120.8603899
##  [126]   884.1133727  -345.1020794   945.7296533   451.8091693  -532.0363467
##  [131]  -215.0660636   136.8797355  -596.6610134  -687.7398353  -484.7143832
##  [136]   -64.9531686    91.0873769  -316.9195820  -343.7177556  -461.5558195
##  [141]  -245.5614974  -268.0170839   102.2926604   138.8762951  -533.1365360
##  [146]  1316.6171261   277.7571261 -1010.2326340   229.7926902  -336.9654193
##  [151]  -615.4206298   -52.5157755  -237.5480154  -365.3760819  -262.9761225
##  [156]  -605.9091454   148.2815919   331.2981813  -233.9405584   197.8609576
##  [161]   336.2655811  -302.7208780   381.3827359   103.3206330   -23.8432616
##  [166]  -576.7468623  -527.8658303  -491.2574650  -218.9896501   355.2569252
##  [171]  -526.5304380  -376.7723301  -299.0235447   788.3041722  -783.1729561
##  [176]  -373.6257871   212.1452083  -424.5300403   503.6110439   184.3641166
##  [181]   470.5864819  -174.5668515   160.7065886    98.1592798  -115.1938966
##  [186]    35.2460536    87.2512931  -391.1530605  -226.9350124    17.5623732
##  [191]   291.0348664  -146.5583549   220.6812707   334.0285127   489.9150333
##  [196]   395.8713919   106.9544720   487.8494309   721.3831296   305.8279838
##  [201]   172.7926571   295.6284770   446.5030224   315.1378834   151.4350432
##  [206]   327.8223400   822.2969403   770.2780229  1151.5571437   560.4421482
##  [211]   946.1113172   806.3252416   907.0857871   826.8964014   381.9663010
##  [216] -2363.3104686 -2430.2910805 -2338.5493138 -2311.5746214 -2245.1922262
##  [221] -2714.7634524 -2013.0233154 -2285.5496832 -2501.3378750 -2014.8037746
##  [226] -2441.8720643 -2287.8391237 -2282.4476716 -2293.5818634 -2010.6227991
##  [231] -2248.1234682 -2418.2587766 -2584.8465143 -2452.3613246 -2253.7413678
##  [236] -2635.5995413 -2493.1530390 -2148.2520070 -2780.8307924 -2443.6445849
##  [241] -2207.7881739  -447.0011693   142.2531229  -231.8365575   113.7479905
##  [246]   583.1296526   610.7232989   -76.8778377   136.7783383   288.1137904
##  [251]  -421.1087484   311.4958086   380.9512607  -113.4070540   566.0004529
##  [256]   336.0074276   337.8071992   528.5825442   367.6848663   223.8189967
##  [261]   185.3828871   502.6171976   629.3956019    -0.4617547   130.8386563
##  [266]   345.7377497   411.7559257   961.9514873   337.8726994   564.4902910
##  [271]   620.9527866   209.9473412   663.2267434   394.5223690  1141.6989759
##  [276]   181.7598460   336.2010606    96.1930697   449.2136201   324.8117364
##  [281]   265.9711154   315.5586131   400.0877090   114.4128323   359.9774417
##  [286]   136.7981590    65.1526795   293.8925060   202.2204487    -4.6449643
##  [291]   276.9210033   522.2540286   181.0365126   137.3386314   209.9665126
##  [296]   -36.2658369   317.5510672   412.9675969   519.3931860   371.3100079
##  [301]   357.3400079   507.4788572   501.8217248   458.2561332   213.2832980
##  [306]   979.9726770   334.9099647    60.2362910   435.1961631   362.9443524
##  [311]   -81.5784487   323.6782362    63.4745923   531.5162179  -362.8363325
##  [316]   384.2975878   128.0892931   405.9864163   419.9243524   454.7450830
##  [321]   256.0091013   121.8491768   675.9358435   106.5148688    12.0128414
##  [326]   267.0218485   519.6665151  -195.4134616   141.8970431   358.6742602
##  [331]  -176.4157398   194.7889244   -92.6901690   590.1318908   752.7572606
##  [336]   500.2497471    98.2719347   310.5843914  1018.8610306   647.0357272
##  [341]  1080.4631153   878.0584345  1038.9744711   925.2799115   967.7830348
##  [346]   796.9804437  1253.2338799  1248.8264802  1156.7768845   992.0259688
##  [351]  1288.2767152  1386.6658177  1205.7503930   962.7190738   929.2646786
##  [356]   769.2268978  1216.4995412   757.9039547   608.4827401   928.7185599
##  [361]  1390.3123175   644.0924063  1249.4742195  1003.9414889   882.1852033
##  [366]   286.9863748  1044.9239912   334.8216483  -561.4433654  -127.3880984
##  [371]   114.5016251   906.2141457  -104.1371511   -53.7951810   526.5592299
##  [376]   -91.8330557  -564.4173795  -637.9615281   149.1544421  -256.3734733
##  [381]  -343.0058934  -211.3798776  1069.9217920  -381.4790183  -107.3450822
##  [386]  -684.2863267   162.4766230   709.2311685   335.6176824   991.1810132
##  [391]  -129.4425733   -89.5247012   941.2128646   416.1724836  -164.4979158
##  [396]  -153.4777174  -368.9902719   348.0242420   -32.7553313  -468.8374633
##  [401]    76.8147119  -195.3397472   208.4084520  -345.5514019  -520.0222238
##  [406]  -189.8932632    46.4782627   -77.6852533  -309.2993604   225.7017081
##  [411]  -300.4507510  -164.1428971   300.2689501   935.2895437  -344.0317556
##  [416]    53.4235138  -546.2506937  -387.3222578   770.1587401   426.6282195
##  [421]   210.6712764  -984.7331935  -197.4895704  -322.7264837   565.7853337
##  [426]   315.4718634  -411.3331802   312.2896326  -102.7591096   154.5223125
##  [431]    18.5882420  -820.3528631    10.7118700  -535.3729951  -155.2761159
##  [436]   185.7233677   253.6864321  -243.6742139  -695.1703600  -483.5787826
##  [441]  1267.3040186  1038.2479206   528.5533021   -29.9759764  -281.0326730
##  [446]  -436.6571338    70.8562129  -191.5849993   773.1979231  -390.2778394
##  [451]  -356.6187253  -541.3677506    -2.9773147   555.1089094   361.8594615
##  [456]  -156.7906066  -561.3514193  -529.0671205  -684.0202869  -311.3767917
##  [461]  -282.2763143   666.8936309   981.3429941   163.1471676  -242.0540470
##  [466]  -393.0072433    68.0627567    33.9008729  -837.2494376   688.9133752
##  [471]  1185.2844619  -153.0505268  -538.8485518  -252.4711894  -692.1905360
##  [476]  -240.2724995  -270.1509013   657.7673362   173.7200028  -294.8902346
##  [481]  -468.0331205   230.6063316  -161.0826090  -322.6724173   365.4214732
##  [486]    21.7728430  -645.0593720   258.4381465   193.3353976  -120.8603899
##  [491]   884.1133727  -345.1020794   945.7296533   451.8091693  -532.0363467
##  [496]  -215.0660636   136.8797355  -596.6610134  -687.7398353  -484.7143832
##  [501]   -64.9531686    91.0873769  -316.9195820  -343.7177556  -461.5558195
##  [506]  -245.5614974  -268.0170839   102.2926604   138.8762951  -533.1365360
##  [511]  1316.6171261   277.7571261 -1010.2326340   229.7926902  -336.9654193
##  [516]  -615.4206298   -52.5157755  -237.5480154  -365.3760819  -262.9761225
##  [521]  -605.9091454   148.2815919   331.2981813  -233.9405584   197.8609576
##  [526]   336.2655811  -302.7208780   381.3827359   103.3206330   -23.8432616
##  [531]  -576.7468623  -527.8658303  -491.2574650  -218.9896501   355.2569252
##  [536]  -526.5304380  -376.7723301  -299.0235447   788.3041722  -783.1729561
##  [541]  -373.6257871   212.1452083  -424.5300403   503.6110439   184.3641166
##  [546]   470.5864819  -174.5668515   160.7065886    98.1592798  -115.1938966
##  [551]    35.2460536    87.2512931  -391.1530605  -226.9350124    17.5623732
##  [556]   291.0348664  -146.5583549   220.6812707   334.0285127   489.9150333
##  [561]   395.8713919   106.9544720   487.8494309   721.3831296   305.8279838
##  [566]   172.7926571   295.6284770   446.5030224   315.1378834   151.4350432
##  [571]   327.8223400   822.2969403   770.2780229  1151.5571437   560.4421482
##  [576]   946.1113172   806.3252416   907.0857871   826.8964014   381.9663010
##  [581] -2363.3104686 -2430.2910805 -2338.5493138 -2311.5746214 -2245.1922262
##  [586] -2714.7634524 -2013.0233154 -2285.5496832 -2501.3378750 -2014.8037746
##  [591] -2441.8720643 -2287.8391237 -2282.4476716 -2293.5818634 -2010.6227991
##  [596] -2248.1234682 -2418.2587766 -2584.8465143 -2452.3613246 -2253.7413678
##  [601] -2635.5995413 -2493.1530390 -2148.2520070 -2780.8307924 -2443.6445849
##  [606] -2207.7881739  -447.0011693   142.2531229  -231.8365575   113.7479905
##  [611]   583.1296526   610.7232989   -76.8778377   136.7783383   288.1137904
##  [616]  -421.1087484   311.4958086   380.9512607  -113.4070540   566.0004529
##  [621]   336.0074276   337.8071992   528.5825442   367.6848663   223.8189967
##  [626]   185.3828871   502.6171976   629.3956019    -0.4617547   130.8386563
##  [631]   345.7377497   411.7559257   961.9514873   337.8726994   564.4902910
##  [636]   620.9527866   209.9473412   663.2267434   394.5223690  1141.6989759
##  [641]   181.7598460   336.2010606    96.1930697   449.2136201   324.8117364
##  [646]   265.9711154   315.5586131   400.0877090   114.4128323   359.9774417
##  [651]   136.7981590    65.1526795   293.8925060   202.2204487    -4.6449643
##  [656]   276.9210033   522.2540286   181.0365126   137.3386314   209.9665126
##  [661]   -36.2658369   317.5510672   412.9675969   519.3931860   371.3100079
##  [666]   357.3400079   507.4788572   501.8217248   458.2561332   213.2832980
##  [671]   979.9726770   334.9099647    60.2362910   435.1961631   362.9443524
##  [676]   -81.5784487   323.6782362    63.4745923   531.5162179  -362.8363325
##  [681]   384.2975878   128.0892931   405.9864163   419.9243524   454.7450830
##  [686]   256.0091013   121.8491768   675.9358435   106.5148688    12.0128414
##  [691]   267.0218485   519.6665151  -195.4134616   141.8970431   358.6742602
##  [696]  -176.4157398   194.7889244   -92.6901690   590.1318908   752.7572606
##  [701]   500.2497471    98.2719347   310.5843914  1018.8610306   647.0357272
##  [706]  1080.4631153   878.0584345  1038.9744711   925.2799115   967.7830348
##  [711]   796.9804437  1253.2338799  1248.8264802  1156.7768845   992.0259688
##  [716]  1288.2767152  1386.6658177  1205.7503930   962.7190738   929.2646786
##  [721]   769.2268978  1216.4995412   757.9039547   608.4827401   928.7185599
##  [726]  1390.3123175   644.0924063  1249.4742195  1003.9414889   882.1852033
##  [731]   286.9863748  1044.9239912   334.8216483  -561.4433654  -127.3880984
##  [736]   114.5016251   906.2141457  -104.1371511   -53.7951810   526.5592299
##  [741]   -91.8330557  -564.4173795  -637.9615281   149.1544421  -256.3734733
##  [746]  -343.0058934  -211.3798776  1069.9217920  -381.4790183  -107.3450822
##  [751]  -684.2863267   162.4766230   709.2311685   335.6176824   991.1810132
##  [756]  -129.4425733   -89.5247012   941.2128646   416.1724836  -164.4979158
##  [761]  -153.4777174  -368.9902719   348.0242420   -32.7553313  -468.8374633
##  [766]    76.8147119  -195.3397472   208.4084520  -345.5514019  -520.0222238
##  [771]  -189.8932632    46.4782627   -77.6852533  -309.2993604   225.7017081
##  [776]  -300.4507510  -164.1428971   300.2689501   935.2895437  -344.0317556
##  [781]    53.4235138  -546.2506937  -387.3222578   770.1587401   426.6282195
##  [786]   210.6712764  -984.7331935  -197.4895704  -322.7264837   565.7853337
##  [791]   315.4718634  -411.3331802   312.2896326  -102.7591096   154.5223125
##  [796]    18.5882420  -820.3528631    10.7118700  -535.3729951  -155.2761159
##  [801]   185.7233677   253.6864321  -243.6742139  -695.1703600  -483.5787826
##  [806]  1267.3040186  1038.2479206   528.5533021   -29.9759764  -281.0326730
##  [811]  -436.6571338    70.8562129  -191.5849993   773.1979231  -390.2778394
##  [816]  -356.6187253  -541.3677506    -2.9773147   555.1089094   361.8594615
##  [821]  -156.7906066  -561.3514193  -529.0671205  -684.0202869  -311.3767917
##  [826]  -282.2763143   666.8936309   981.3429941   163.1471676  -242.0540470
##  [831]  -393.0072433    68.0627567    33.9008729  -837.2494376   688.9133752
##  [836]  1185.2844619  -153.0505268  -538.8485518  -252.4711894  -692.1905360
##  [841]  -240.2724995  -270.1509013   657.7673362   173.7200028  -294.8902346
##  [846]  -468.0331205   230.6063316  -161.0826090  -322.6724173   365.4214732
##  [851]    21.7728430  -645.0593720   258.4381465   193.3353976  -120.8603899
##  [856]   884.1133727  -345.1020794   945.7296533   451.8091693  -532.0363467
##  [861]  -215.0660636   136.8797355  -596.6610134  -687.7398353  -484.7143832
##  [866]   -64.9531686    91.0873769  -316.9195820  -343.7177556  -461.5558195
##  [871]  -245.5614974  -268.0170839   102.2926604   138.8762951  -533.1365360
##  [876]  1316.6171261   277.7571261 -1010.2326340   229.7926902  -336.9654193
##  [881]  -615.4206298   -52.5157755  -237.5480154  -365.3760819  -262.9761225
##  [886]  -605.9091454   148.2815919   331.2981813  -233.9405584   197.8609576
##  [891]   336.2655811  -302.7208780   381.3827359   103.3206330   -23.8432616
##  [896]  -576.7468623  -527.8658303  -491.2574650  -218.9896501   355.2569252
##  [901]  -526.5304380  -376.7723301  -299.0235447   788.3041722  -783.1729561
##  [906]  -373.6257871   212.1452083  -424.5300403   503.6110439   184.3641166
##  [911]   470.5864819  -174.5668515   160.7065886    98.1592798  -115.1938966
##  [916]    35.2460536    87.2512931  -391.1530605  -226.9350124    17.5623732
##  [921]   291.0348664  -146.5583549   220.6812707   334.0285127   489.9150333
##  [926]   395.8713919   106.9544720   487.8494309   721.3831296   305.8279838
##  [931]   172.7926571   295.6284770   446.5030224   315.1378834   151.4350432
##  [936]   327.8223400   822.2969403   770.2780229  1151.5571437   560.4421482
##  [941]   946.1113172   806.3252416   907.0857871   826.8964014   381.9663010
##  [946] -2363.3104686 -2430.2910805 -2338.5493138 -2311.5746214 -2245.1922262
##  [951] -2714.7634524 -2013.0233154 -2285.5496832 -2501.3378750 -2014.8037746
##  [956] -2441.8720643 -2287.8391237 -2282.4476716 -2293.5818634 -2010.6227991
##  [961] -2248.1234682 -2418.2587766 -2584.8465143 -2452.3613246 -2253.7413678
##  [966] -2635.5995413 -2493.1530390 -2148.2520070 -2780.8307924 -2443.6445849
##  [971] -2207.7881739  -447.0011693   142.2531229  -231.8365575   113.7479905
##  [976]   583.1296526   610.7232989   -76.8778377   136.7783383   288.1137904
##  [981]  -421.1087484   311.4958086   380.9512607  -113.4070540   566.0004529
##  [986]   336.0074276   337.8071992   528.5825442   367.6848663   223.8189967
##  [991]   185.3828871   502.6171976   629.3956019    -0.4617547   130.8386563
##  [996]   345.7377497   411.7559257   961.9514873   337.8726994   564.4902910
## [1001]   620.9527866   209.9473412   663.2267434   394.5223690  1141.6989759
## [1006]   181.7598460   336.2010606    96.1930697   449.2136201   324.8117364
## [1011]   265.9711154   315.5586131   400.0877090   114.4128323   359.9774417
## [1016]   136.7981590    65.1526795   293.8925060   202.2204487    -4.6449643
## [1021]   276.9210033   522.2540286   181.0365126   137.3386314   209.9665126
## [1026]   -36.2658369   317.5510672   412.9675969   519.3931860   371.3100079
## [1031]   357.3400079   507.4788572   501.8217248   458.2561332   213.2832980
## [1036]   979.9726770   334.9099647    60.2362910   435.1961631   362.9443524
## [1041]   -81.5784487   323.6782362    63.4745923   531.5162179  -362.8363325
## [1046]   384.2975878   128.0892931   405.9864163   419.9243524   454.7450830
## [1051]   256.0091013   121.8491768   675.9358435   106.5148688    12.0128414
## [1056]   267.0218485   519.6665151  -195.4134616   141.8970431   358.6742602
## [1061]  -176.4157398   194.7889244   -92.6901690   590.1318908   752.7572606
## [1066]   500.2497471    98.2719347   310.5843914  1018.8610306   647.0357272
## [1071]  1080.4631153   878.0584345  1038.9744711   925.2799115   967.7830348
## [1076]   796.9804437  1253.2338799  1248.8264802  1156.7768845   992.0259688
## [1081]  1288.2767152  1386.6658177  1205.7503930   962.7190738   929.2646786
## [1086]   769.2268978  1216.4995412   757.9039547   608.4827401   928.7185599
## [1091]  1390.3123175   644.0924063  1249.4742195  1003.9414889   882.1852033
## [1096]   286.9863748  1044.9239912   334.8216483  -561.4433654  -127.3880984
## [1101]   114.5016251   906.2141457  -104.1371511   -53.7951810   526.5592299
## [1106]   -91.8330557  -564.4173795  -637.9615281   149.1544421  -256.3734733
## [1111]  -343.0058934  -211.3798776  1069.9217920  -381.4790183  -107.3450822
## [1116]  -684.2863267   162.4766230   709.2311685   335.6176824   991.1810132
## [1121]  -129.4425733   -89.5247012   941.2128646   416.1724836  -164.4979158
## [1126]  -153.4777174  -368.9902719   348.0242420   -32.7553313  -468.8374633
## [1131]    76.8147119  -195.3397472   208.4084520  -345.5514019  -520.0222238
## [1136]  -189.8932632    46.4782627   -77.6852533  -309.2993604   225.7017081
## [1141]  -300.4507510  -164.1428971   300.2689501   935.2895437  -344.0317556
## [1146]    53.4235138  -546.2506937  -387.3222578   770.1587401   426.6282195
## [1151]   210.6712764  -984.7331935  -197.4895704  -322.7264837   565.7853337
## [1156]   315.4718634  -411.3331802   312.2896326  -102.7591096   154.5223125
## [1161]    18.5882420  -820.3528631    10.7118700  -535.3729951  -155.2761159
## [1166]   185.7233677   253.6864321  -243.6742139  -695.1703600  -483.5787826
## [1171]  1267.3040186  1038.2479206   528.5533021   -29.9759764  -281.0326730
## [1176]  -436.6571338    70.8562129  -191.5849993   773.1979231  -390.2778394
## [1181]  -356.6187253  -541.3677506    -2.9773147   555.1089094   361.8594615
## [1186]  -156.7906066  -561.3514193  -529.0671205  -684.0202869  -311.3767917
## [1191]  -282.2763143   666.8936309   981.3429941   163.1471676  -242.0540470
## [1196]  -393.0072433    68.0627567    33.9008729  -837.2494376   688.9133752
## [1201]  1185.2844619  -153.0505268  -538.8485518  -252.4711894  -692.1905360
## [1206]  -240.2724995  -270.1509013   657.7673362   173.7200028  -294.8902346
## [1211]  -468.0331205   230.6063316  -161.0826090  -322.6724173   365.4214732
## [1216]    21.7728430  -645.0593720   258.4381465   193.3353976  -120.8603899
## [1221]   884.1133727  -345.1020794   945.7296533   451.8091693  -532.0363467
## [1226]  -215.0660636   136.8797355  -596.6610134  -687.7398353  -484.7143832
## [1231]   -64.9531686    91.0873769  -316.9195820  -343.7177556  -461.5558195
## [1236]  -245.5614974  -268.0170839   102.2926604   138.8762951  -533.1365360
## [1241]  1316.6171261   277.7571261 -1010.2326340   229.7926902  -336.9654193
## [1246]  -615.4206298   -52.5157755  -237.5480154  -365.3760819  -262.9761225
## [1251]  -605.9091454   148.2815919   331.2981813  -233.9405584   197.8609576
## [1256]   336.2655811  -302.7208780   381.3827359   103.3206330   -23.8432616
## [1261]  -576.7468623  -527.8658303  -491.2574650  -218.9896501   355.2569252
## [1266]  -526.5304380  -376.7723301  -299.0235447   788.3041722  -783.1729561
## [1271]  -373.6257871   212.1452083  -424.5300403   503.6110439   184.3641166
## [1276]   470.5864819  -174.5668515   160.7065886    98.1592798  -115.1938966
## [1281]    35.2460536    87.2512931  -391.1530605  -226.9350124    17.5623732
## [1286]   291.0348664  -146.5583549   220.6812707   334.0285127   489.9150333
## [1291]   395.8713919   106.9544720   487.8494309   721.3831296   305.8279838
## [1296]   172.7926571   295.6284770   446.5030224   315.1378834   151.4350432
## [1301]   327.8223400   822.2969403   770.2780229  1151.5571437   560.4421482
## [1306]   946.1113172   806.3252416   907.0857871   826.8964014   381.9663010
## [1311] -2363.3104686 -2430.2910805 -2338.5493138 -2311.5746214 -2245.1922262
## [1316] -2714.7634524 -2013.0233154 -2285.5496832 -2501.3378750 -2014.8037746
## [1321] -2441.8720643 -2287.8391237 -2282.4476716 -2293.5818634 -2010.6227991
## [1326] -2248.1234682 -2418.2587766 -2584.8465143 -2452.3613246 -2253.7413678
## [1331] -2635.5995413 -2493.1530390 -2148.2520070 -2780.8307924 -2443.6445849
## [1336] -2207.7881739  -447.0011693   142.2531229  -231.8365575   113.7479905
## [1341]   583.1296526   610.7232989   -76.8778377   136.7783383   288.1137904
## [1346]  -421.1087484   311.4958086   380.9512607  -113.4070540   566.0004529
## [1351]   336.0074276   337.8071992   528.5825442   367.6848663   223.8189967
## [1356]   185.3828871   502.6171976   629.3956019    -0.4617547   130.8386563
## [1361]   345.7377497   411.7559257   961.9514873   337.8726994   564.4902910
## [1366]   620.9527866   209.9473412   663.2267434   394.5223690  1141.6989759
## [1371]   181.7598460   336.2010606    96.1930697   449.2136201   324.8117364
## [1376]   265.9711154   315.5586131   400.0877090   114.4128323   359.9774417
## [1381]   136.7981590    65.1526795   293.8925060   202.2204487    -4.6449643
## [1386]   276.9210033   522.2540286   181.0365126   137.3386314   209.9665126
## [1391]   -36.2658369   317.5510672   412.9675969   519.3931860   371.3100079
## [1396]   357.3400079   507.4788572   501.8217248   458.2561332   213.2832980
## [1401]   979.9726770   334.9099647    60.2362910   435.1961631   362.9443524
## [1406]   -81.5784487   323.6782362    63.4745923   531.5162179  -362.8363325
## [1411]   384.2975878   128.0892931   405.9864163   419.9243524   454.7450830
## [1416]   256.0091013   121.8491768   675.9358435   106.5148688    12.0128414
## [1421]   267.0218485   519.6665151  -195.4134616   141.8970431   358.6742602
## [1426]  -176.4157398   194.7889244   -92.6901690   590.1318908   752.7572606
## [1431]   500.2497471    98.2719347   310.5843914  1018.8610306   647.0357272
## [1436]  1080.4631153   878.0584345  1038.9744711   925.2799115   967.7830348
## [1441]   796.9804437  1253.2338799  1248.8264802  1156.7768845   992.0259688
## [1446]  1288.2767152  1386.6658177  1205.7503930   962.7190738   929.2646786
## [1451]   769.2268978  1216.4995412   757.9039547   608.4827401   928.7185599
## [1456]  1390.3123175   644.0924063  1249.4742195  1003.9414889   882.1852033
## [1461]   286.9863748  1044.9239912   334.8216483  -561.4433654  -127.3880984
## [1466]   114.5016251   906.2141457  -104.1371511   -53.7951810   526.5592299
## [1471]   -91.8330557  -564.4173795  -637.9615281   149.1544421  -256.3734733
## [1476]  -343.0058934  -211.3798776  1069.9217920  -381.4790183  -107.3450822
## [1481]  -684.2863267   162.4766230   709.2311685   335.6176824   991.1810132
## [1486]  -129.4425733   -89.5247012   941.2128646   416.1724836  -164.4979158
## [1491]  -153.4777174  -368.9902719   348.0242420   -32.7553313  -468.8374633
## [1496]    76.8147119  -195.3397472   208.4084520  -345.5514019  -520.0222238
## [1501]  -189.8932632    46.4782627   -77.6852533  -309.2993604   225.7017081
## [1506]  -300.4507510  -164.1428971   300.2689501   935.2895437  -344.0317556
## [1511]    53.4235138  -546.2506937  -387.3222578   770.1587401   426.6282195
## [1516]   210.6712764  -984.7331935  -197.4895704  -322.7264837   565.7853337
## [1521]   315.4718634  -411.3331802   312.2896326  -102.7591096   154.5223125
## [1526]    18.5882420  -820.3528631    10.7118700  -535.3729951  -155.2761159
## [1531]   185.7233677   253.6864321  -243.6742139  -695.1703600  -483.5787826
## [1536]  1267.3040186  1038.2479206   528.5533021   -29.9759764  -281.0326730
## [1541]  -436.6571338    70.8562129  -191.5849993   773.1979231  -390.2778394
## [1546]  -356.6187253  -541.3677506    -2.9773147   555.1089094   361.8594615
## [1551]  -156.7906066  -561.3514193  -529.0671205  -684.0202869  -311.3767917
## [1556]  -282.2763143   666.8936309   981.3429941   163.1471676  -242.0540470
## [1561]  -393.0072433    68.0627567    33.9008729  -837.2494376   688.9133752
## [1566]  1185.2844619  -153.0505268  -538.8485518  -252.4711894  -692.1905360
## [1571]  -240.2724995  -270.1509013   657.7673362   173.7200028  -294.8902346
## [1576]  -468.0331205   230.6063316  -161.0826090  -322.6724173   365.4214732
## [1581]    21.7728430  -645.0593720   258.4381465   193.3353976  -120.8603899
## [1586]   884.1133727  -345.1020794   945.7296533   451.8091693  -532.0363467
## [1591]  -215.0660636   136.8797355  -596.6610134  -687.7398353  -484.7143832
## [1596]   -64.9531686    91.0873769  -316.9195820  -343.7177556  -461.5558195
## [1601]  -245.5614974  -268.0170839   102.2926604   138.8762951  -533.1365360
## [1606]  1316.6171261   277.7571261 -1010.2326340   229.7926902  -336.9654193
## [1611]  -615.4206298   -52.5157755  -237.5480154  -365.3760819  -262.9761225
## [1616]  -605.9091454   148.2815919   331.2981813  -233.9405584   197.8609576
## [1621]   336.2655811  -302.7208780   381.3827359   103.3206330   -23.8432616
## [1626]  -576.7468623  -527.8658303  -491.2574650  -218.9896501   355.2569252
## [1631]  -526.5304380  -376.7723301  -299.0235447   788.3041722  -783.1729561
## [1636]  -373.6257871   212.1452083  -424.5300403   503.6110439   184.3641166
## [1641]   470.5864819  -174.5668515   160.7065886    98.1592798  -115.1938966
## [1646]    35.2460536    87.2512931  -391.1530605  -226.9350124    17.5623732
## [1651]   291.0348664  -146.5583549   220.6812707   334.0285127   489.9150333
## [1656]   395.8713919   106.9544720   487.8494309   721.3831296   305.8279838
## [1661]   172.7926571   295.6284770   446.5030224   315.1378834   151.4350432
## [1666]   327.8223400   822.2969403   770.2780229  1151.5571437   560.4421482
## [1671]   946.1113172   806.3252416   907.0857871   826.8964014   381.9663010
## [1676] -2363.3104686 -2430.2910805 -2338.5493138 -2311.5746214 -2245.1922262
## [1681] -2714.7634524 -2013.0233154 -2285.5496832 -2501.3378750 -2014.8037746
## [1686] -2441.8720643 -2287.8391237 -2282.4476716 -2293.5818634 -2010.6227991
## [1691] -2248.1234682 -2418.2587766 -2584.8465143 -2452.3613246 -2253.7413678
## [1696] -2635.5995413 -2493.1530390 -2148.2520070 -2780.8307924 -2443.6445849
## [1701] -2207.7881739  -447.0011693   142.2531229  -231.8365575   113.7479905
## [1706]   583.1296526   610.7232989   -76.8778377   136.7783383   288.1137904
## [1711]  -421.1087484   311.4958086   380.9512607  -113.4070540   566.0004529
## [1716]   336.0074276   337.8071992   528.5825442   367.6848663   223.8189967
## [1721]   185.3828871   502.6171976   629.3956019    -0.4617547   130.8386563
## [1726]   345.7377497   411.7559257   961.9514873   337.8726994   564.4902910
## [1731]   620.9527866   209.9473412   663.2267434   394.5223690  1141.6989759
## [1736]   181.7598460   336.2010606    96.1930697   449.2136201   324.8117364
## [1741]   265.9711154   315.5586131   400.0877090   114.4128323   359.9774417
## [1746]   136.7981590    65.1526795   293.8925060   202.2204487    -4.6449643
## [1751]   276.9210033   522.2540286   181.0365126   137.3386314   209.9665126
## [1756]   -36.2658369   317.5510672   412.9675969   519.3931860   371.3100079
## [1761]   357.3400079   507.4788572   501.8217248   458.2561332   213.2832980
## [1766]   979.9726770   334.9099647    60.2362910   435.1961631   362.9443524
## [1771]   -81.5784487   323.6782362    63.4745923   531.5162179  -362.8363325
## [1776]   384.2975878   128.0892931   405.9864163   419.9243524   454.7450830
## [1781]   256.0091013   121.8491768   675.9358435   106.5148688    12.0128414
## [1786]   267.0218485   519.6665151  -195.4134616   141.8970431   358.6742602
## [1791]  -176.4157398   194.7889244   -92.6901690   590.1318908   752.7572606
## [1796]   500.2497471    98.2719347   310.5843914  1018.8610306   647.0357272
## [1801]  1080.4631153   878.0584345  1038.9744711   925.2799115   967.7830348
## [1806]   796.9804437  1253.2338799  1248.8264802  1156.7768845   992.0259688
## [1811]  1288.2767152  1386.6658177  1205.7503930   962.7190738   929.2646786
## [1816]   769.2268978  1216.4995412   757.9039547   608.4827401   928.7185599
## [1821]  1390.3123175   644.0924063  1249.4742195  1003.9414889   882.1852033
## [1826]   286.9863748  1044.9239912   334.8216483  -561.4433654  -127.3880984
## [1831]   114.5016251   906.2141457  -104.1371511   -53.7951810   526.5592299
## [1836]   -91.8330557  -564.4173795  -637.9615281   149.1544421  -256.3734733
## [1841]  -343.0058934  -211.3798776  1069.9217920  -381.4790183  -107.3450822
## [1846]  -684.2863267   162.4766230   709.2311685   335.6176824   991.1810132
## [1851]  -129.4425733   -89.5247012   941.2128646   416.1724836  -164.4979158
## [1856]  -153.4777174  -368.9902719   348.0242420   -32.7553313  -468.8374633
## [1861]    76.8147119  -195.3397472   208.4084520  -345.5514019  -520.0222238
## [1866]  -189.8932632    46.4782627   -77.6852533  -309.2993604   225.7017081
## [1871]  -300.4507510  -164.1428971   300.2689501   935.2895437  -344.0317556
## [1876]    53.4235138  -546.2506937  -387.3222578   770.1587401   426.6282195
## [1881]   210.6712764  -984.7331935  -197.4895704  -322.7264837   565.7853337
## [1886]   315.4718634  -411.3331802   312.2896326  -102.7591096   154.5223125
## [1891]    18.5882420  -820.3528631    10.7118700  -535.3729951  -155.2761159
## [1896]   185.7233677   253.6864321  -243.6742139  -695.1703600  -483.5787826
## [1901]  1267.3040186  1038.2479206   528.5533021   -29.9759764  -281.0326730
## [1906]  -436.6571338    70.8562129  -191.5849993   773.1979231  -390.2778394
## [1911]  -356.6187253  -541.3677506    -2.9773147   555.1089094   361.8594615
## [1916]  -156.7906066  -561.3514193  -529.0671205  -684.0202869  -311.3767917
## [1921]  -282.2763143   666.8936309   981.3429941   163.1471676  -242.0540470
## [1926]  -393.0072433    68.0627567    33.9008729  -837.2494376   688.9133752
## [1931]  1185.2844619  -153.0505268  -538.8485518  -252.4711894  -692.1905360
## [1936]  -240.2724995  -270.1509013   657.7673362   173.7200028  -294.8902346
## [1941]  -468.0331205   230.6063316  -161.0826090  -322.6724173   365.4214732
## [1946]    21.7728430  -645.0593720   258.4381465   193.3353976  -120.8603899
## [1951]   884.1133727  -345.1020794   945.7296533   451.8091693  -532.0363467
## [1956]  -215.0660636   136.8797355  -596.6610134  -687.7398353  -484.7143832
## [1961]   -64.9531686    91.0873769  -316.9195820  -343.7177556  -461.5558195
## [1966]  -245.5614974  -268.0170839   102.2926604   138.8762951  -533.1365360
## [1971]  1316.6171261   277.7571261 -1010.2326340   229.7926902  -336.9654193
## [1976]  -615.4206298   -52.5157755  -237.5480154  -365.3760819  -262.9761225
## [1981]  -605.9091454   148.2815919   331.2981813  -233.9405584   197.8609576
## [1986]   336.2655811  -302.7208780   381.3827359   103.3206330   -23.8432616
## [1991]  -576.7468623  -527.8658303  -491.2574650  -218.9896501   355.2569252
## [1996]  -526.5304380  -376.7723301  -299.0235447   788.3041722  -783.1729561
## [2001]  -373.6257871   212.1452083  -424.5300403   503.6110439   184.3641166
## [2006]   470.5864819  -174.5668515   160.7065886    98.1592798  -115.1938966
## [2011]    35.2460536    87.2512931  -391.1530605  -226.9350124    17.5623732
## [2016]   291.0348664  -146.5583549   220.6812707   334.0285127   489.9150333
## [2021]   395.8713919   106.9544720   487.8494309   721.3831296   305.8279838
## [2026]   172.7926571   295.6284770   446.5030224   315.1378834   151.4350432
## [2031]   327.8223400   822.2969403   770.2780229  1151.5571437   560.4421482
## [2036]   946.1113172   806.3252416   907.0857871   826.8964014   381.9663010
## [2041] -2363.3104686 -2430.2910805 -2338.5493138 -2311.5746214 -2245.1922262
## [2046] -2714.7634524 -2013.0233154 -2285.5496832 -2501.3378750 -2014.8037746
## [2051] -2441.8720643 -2287.8391237 -2282.4476716 -2293.5818634 -2010.6227991
## [2056] -2248.1234682 -2418.2587766 -2584.8465143 -2452.3613246 -2253.7413678
## [2061] -2635.5995413 -2493.1530390 -2148.2520070 -2780.8307924 -2443.6445849
## [2066] -2207.7881739  -447.0011693   142.2531229  -231.8365575   113.7479905
## [2071]   583.1296526   610.7232989   -76.8778377   136.7783383   288.1137904
## [2076]  -421.1087484   311.4958086   380.9512607  -113.4070540   566.0004529
## [2081]   336.0074276   337.8071992   528.5825442   367.6848663   223.8189967
## [2086]   185.3828871   502.6171976   629.3956019    -0.4617547   130.8386563
## [2091]   345.7377497   411.7559257   961.9514873   337.8726994   564.4902910
## [2096]   620.9527866   209.9473412   663.2267434   394.5223690  1141.6989759
## [2101]   181.7598460   336.2010606    96.1930697   449.2136201   324.8117364
## [2106]   265.9711154   315.5586131   400.0877090   114.4128323   359.9774417
## [2111]   136.7981590    65.1526795   293.8925060   202.2204487    -4.6449643
## [2116]   276.9210033   522.2540286   181.0365126   137.3386314   209.9665126
## [2121]   -36.2658369   317.5510672   412.9675969   519.3931860   371.3100079
## [2126]   357.3400079   507.4788572   501.8217248   458.2561332   213.2832980
## [2131]   979.9726770   334.9099647    60.2362910   435.1961631   362.9443524
## [2136]   -81.5784487   323.6782362    63.4745923   531.5162179  -362.8363325
## [2141]   384.2975878   128.0892931   405.9864163   419.9243524   454.7450830
## [2146]   256.0091013   121.8491768   675.9358435   106.5148688    12.0128414
## [2151]   267.0218485   519.6665151  -195.4134616   141.8970431   358.6742602
## [2156]  -176.4157398   194.7889244   -92.6901690   590.1318908   752.7572606
## [2161]   500.2497471    98.2719347   310.5843914  1018.8610306   647.0357272
## [2166]  1080.4631153   878.0584345  1038.9744711   925.2799115   967.7830348
## [2171]   796.9804437  1253.2338799  1248.8264802  1156.7768845   992.0259688
## [2176]  1288.2767152  1386.6658177  1205.7503930   962.7190738   929.2646786
## [2181]   769.2268978  1216.4995412   757.9039547   608.4827401   928.7185599
## [2186]  1390.3123175   644.0924063  1249.4742195  1003.9414889   882.1852033
## 
## $trend
## Time Series:
## Start = c(1, 1) 
## End = c(2, 365) 
## Frequency = 365 
##           [,1]     [,2]     [,3]
## 1.000000    NA       NA       NA
## 1.002740    NA       NA       NA
## 1.005479    NA       NA       NA
## 1.008219    NA       NA       NA
## 1.010959    NA       NA       NA
## 1.013699    NA       NA       NA
## 1.016438    NA       NA       NA
## 1.019178    NA       NA       NA
## 1.021918    NA       NA       NA
## 1.024658    NA       NA       NA
## 1.027397    NA       NA       NA
## 1.030137    NA       NA       NA
## 1.032877    NA       NA       NA
## 1.035616    NA       NA       NA
## 1.038356    NA       NA       NA
## 1.041096    NA       NA       NA
## 1.043836    NA       NA       NA
## 1.046575    NA       NA       NA
## 1.049315    NA       NA       NA
## 1.052055    NA       NA       NA
## 1.054795    NA       NA       NA
## 1.057534    NA       NA       NA
## 1.060274    NA       NA       NA
## 1.063014    NA       NA       NA
## 1.065753    NA       NA       NA
## 1.068493    NA       NA       NA
## 1.071233    NA       NA       NA
## 1.073973    NA       NA       NA
## 1.076712    NA       NA       NA
## 1.079452    NA       NA       NA
## 1.082192    NA       NA       NA
## 1.084932    NA       NA       NA
## 1.087671    NA       NA       NA
## 1.090411    NA       NA       NA
## 1.093151    NA       NA       NA
## 1.095890    NA       NA       NA
## 1.098630    NA       NA       NA
## 1.101370    NA       NA       NA
## 1.104110    NA       NA       NA
## 1.106849    NA       NA       NA
## 1.109589    NA       NA       NA
## 1.112329    NA       NA       NA
## 1.115068    NA       NA       NA
## 1.117808    NA       NA       NA
## 1.120548    NA       NA       NA
## 1.123288    NA       NA       NA
## 1.126027    NA       NA       NA
## 1.128767    NA       NA       NA
## 1.131507    NA       NA       NA
## 1.134247    NA       NA       NA
## 1.136986    NA       NA       NA
## 1.139726    NA       NA       NA
## 1.142466    NA       NA       NA
## 1.145205    NA       NA       NA
## 1.147945    NA       NA       NA
## 1.150685    NA       NA       NA
## 1.153425    NA       NA       NA
## 1.156164    NA       NA       NA
## 1.158904    NA       NA       NA
## 1.161644    NA       NA       NA
## 1.164384    NA       NA       NA
## 1.167123    NA       NA       NA
## 1.169863    NA       NA       NA
## 1.172603    NA       NA       NA
## 1.175342    NA       NA       NA
## 1.178082    NA       NA       NA
## 1.180822    NA       NA       NA
## 1.183562    NA       NA       NA
## 1.186301    NA       NA       NA
## 1.189041    NA       NA       NA
## 1.191781    NA       NA       NA
## 1.194521    NA       NA       NA
## 1.197260    NA       NA       NA
## 1.200000    NA       NA       NA
## 1.202740    NA       NA       NA
## 1.205479    NA       NA       NA
## 1.208219    NA       NA       NA
## 1.210959    NA       NA       NA
## 1.213699    NA       NA       NA
## 1.216438    NA       NA       NA
## 1.219178    NA       NA       NA
## 1.221918    NA       NA       NA
## 1.224658    NA       NA       NA
## 1.227397    NA       NA       NA
## 1.230137    NA       NA       NA
## 1.232877    NA       NA       NA
## 1.235616    NA       NA       NA
## 1.238356    NA       NA       NA
## 1.241096    NA       NA       NA
## 1.243836    NA       NA       NA
## 1.246575    NA       NA       NA
## 1.249315    NA       NA       NA
## 1.252055    NA       NA       NA
## 1.254795    NA       NA       NA
## 1.257534    NA       NA       NA
## 1.260274    NA       NA       NA
## 1.263014    NA       NA       NA
## 1.265753    NA       NA       NA
## 1.268493    NA       NA       NA
## 1.271233    NA       NA       NA
## 1.273973    NA       NA       NA
## 1.276712    NA       NA       NA
## 1.279452    NA       NA       NA
## 1.282192    NA       NA       NA
## 1.284932    NA       NA       NA
## 1.287671    NA       NA       NA
## 1.290411    NA       NA       NA
## 1.293151    NA       NA       NA
## 1.295890    NA       NA       NA
## 1.298630    NA       NA       NA
## 1.301370    NA       NA       NA
## 1.304110    NA       NA       NA
## 1.306849    NA       NA       NA
## 1.309589    NA       NA       NA
## 1.312329    NA       NA       NA
## 1.315068    NA       NA       NA
## 1.317808    NA       NA       NA
## 1.320548    NA       NA       NA
## 1.323288    NA       NA       NA
## 1.326027    NA       NA       NA
## 1.328767    NA       NA       NA
## 1.331507    NA       NA       NA
## 1.334247    NA       NA       NA
## 1.336986    NA       NA       NA
## 1.339726    NA       NA       NA
## 1.342466    NA       NA       NA
## 1.345205    NA       NA       NA
## 1.347945    NA       NA       NA
## 1.350685    NA       NA       NA
## 1.353425    NA       NA       NA
## 1.356164    NA       NA       NA
## 1.358904    NA       NA       NA
## 1.361644    NA       NA       NA
## 1.364384    NA       NA       NA
## 1.367123    NA       NA       NA
## 1.369863    NA       NA       NA
## 1.372603    NA       NA       NA
## 1.375342    NA       NA       NA
## 1.378082    NA       NA       NA
## 1.380822    NA       NA       NA
## 1.383562    NA       NA       NA
## 1.386301    NA       NA       NA
## 1.389041    NA       NA       NA
## 1.391781    NA       NA       NA
## 1.394521    NA       NA       NA
## 1.397260    NA       NA       NA
## 1.400000    NA       NA       NA
## 1.402740    NA       NA       NA
## 1.405479    NA       NA       NA
## 1.408219    NA       NA       NA
## 1.410959    NA       NA       NA
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## 2.501370    NA       NA       NA
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## 2.690411    NA       NA       NA
## 2.693151    NA       NA       NA
## 2.695890    NA       NA       NA
## 2.698630    NA       NA       NA
## 2.701370    NA       NA       NA
## 2.704110    NA       NA       NA
## 2.706849    NA       NA       NA
## 2.709589    NA       NA       NA
## 2.712329    NA       NA       NA
## 2.715068    NA       NA       NA
## 2.717808    NA       NA       NA
## 2.720548    NA       NA       NA
## 2.723288    NA       NA       NA
## 2.726027    NA       NA       NA
## 2.728767    NA       NA       NA
## 2.731507    NA       NA       NA
## 2.734247    NA       NA       NA
## 2.736986    NA       NA       NA
## 2.739726    NA       NA       NA
## 2.742466    NA       NA       NA
## 2.745205    NA       NA       NA
## 2.747945    NA       NA       NA
## 2.750685    NA       NA       NA
## 2.753425    NA       NA       NA
## 2.756164    NA       NA       NA
## 2.758904    NA       NA       NA
## 2.761644    NA       NA       NA
## 2.764384    NA       NA       NA
## 2.767123    NA       NA       NA
## 2.769863    NA       NA       NA
## 2.772603    NA       NA       NA
## 2.775342    NA       NA       NA
## 2.778082    NA       NA       NA
## 2.780822    NA       NA       NA
## 2.783562    NA       NA       NA
## 2.786301    NA       NA       NA
## 2.789041    NA       NA       NA
## 2.791781    NA       NA       NA
## 2.794521    NA       NA       NA
## 2.797260    NA       NA       NA
## 2.800000    NA       NA       NA
## 2.802740    NA       NA       NA
## 2.805479    NA       NA       NA
## 2.808219    NA       NA       NA
## 2.810959    NA       NA       NA
## 2.813699    NA       NA       NA
## 2.816438    NA       NA       NA
## 2.819178    NA       NA       NA
## 2.821918    NA       NA       NA
## 2.824658    NA       NA       NA
## 2.827397    NA       NA       NA
## 2.830137    NA       NA       NA
## 2.832877    NA       NA       NA
## 2.835616    NA       NA       NA
## 2.838356    NA       NA       NA
## 2.841096    NA       NA       NA
## 2.843836    NA       NA       NA
## 2.846575    NA       NA       NA
## 2.849315    NA       NA       NA
## 2.852055    NA       NA       NA
## 2.854795    NA       NA       NA
## 2.857534    NA       NA       NA
## 2.860274    NA       NA       NA
## 2.863014    NA       NA       NA
## 2.865753    NA       NA       NA
## 2.868493    NA       NA       NA
## 2.871233    NA       NA       NA
## 2.873973    NA       NA       NA
## 2.876712    NA       NA       NA
## 2.879452    NA       NA       NA
## 2.882192    NA       NA       NA
## 2.884932    NA       NA       NA
## 2.887671    NA       NA       NA
## 2.890411    NA       NA       NA
## 2.893151    NA       NA       NA
## 2.895890    NA       NA       NA
## 2.898630    NA       NA       NA
## 2.901370    NA       NA       NA
## 2.904110    NA       NA       NA
## 2.906849    NA       NA       NA
## 2.909589    NA       NA       NA
## 2.912329    NA       NA       NA
## 2.915068    NA       NA       NA
## 2.917808    NA       NA       NA
## 2.920548    NA       NA       NA
## 2.923288    NA       NA       NA
## 2.926027    NA       NA       NA
## 2.928767    NA       NA       NA
## 2.931507    NA       NA       NA
## 2.934247    NA       NA       NA
## 2.936986    NA       NA       NA
## 2.939726    NA       NA       NA
## 2.942466    NA       NA       NA
## 2.945205    NA       NA       NA
## 2.947945    NA       NA       NA
## 2.950685    NA       NA       NA
## 2.953425    NA       NA       NA
## 2.956164    NA       NA       NA
## 2.958904    NA       NA       NA
## 2.961644    NA       NA       NA
## 2.964384    NA       NA       NA
## 2.967123    NA       NA       NA
## 2.969863    NA       NA       NA
## 2.972603    NA       NA       NA
## 2.975342    NA       NA       NA
## 2.978082    NA       NA       NA
## 2.980822    NA       NA       NA
## 2.983562    NA       NA       NA
## 2.986301    NA       NA       NA
## 2.989041    NA       NA       NA
## 2.991781    NA       NA       NA
## 2.994521    NA       NA       NA
## 2.997260    NA       NA       NA
## 
## $random
## Time Series:
## Start = c(1, 1) 
## End = c(2, 365) 
## Frequency = 365 
##          x - seasonal.x.date x - seasonal.x.ca x - seasonal.x.nb_ventes
## 1.000000                  NA                NA                       NA
## 1.002740                  NA                NA                       NA
## 1.005479                  NA                NA                       NA
## 1.008219                  NA                NA                       NA
## 1.010959                  NA                NA                       NA
## 1.013699                  NA                NA                       NA
## 1.016438                  NA                NA                       NA
## 1.019178                  NA                NA                       NA
## 1.021918                  NA                NA                       NA
## 1.024658                  NA                NA                       NA
## 1.027397                  NA                NA                       NA
## 1.030137                  NA                NA                       NA
## 1.032877                  NA                NA                       NA
## 1.035616                  NA                NA                       NA
## 1.038356                  NA                NA                       NA
## 1.041096                  NA                NA                       NA
## 1.043836                  NA                NA                       NA
## 1.046575                  NA                NA                       NA
## 1.049315                  NA                NA                       NA
## 1.052055                  NA                NA                       NA
## 1.054795                  NA                NA                       NA
## 1.057534                  NA                NA                       NA
## 1.060274                  NA                NA                       NA
## 1.063014                  NA                NA                       NA
## 1.065753                  NA                NA                       NA
## 1.068493                  NA                NA                       NA
## 1.071233                  NA                NA                       NA
## 1.073973                  NA                NA                       NA
## 1.076712                  NA                NA                       NA
## 1.079452                  NA                NA                       NA
## 1.082192                  NA                NA                       NA
## 1.084932                  NA                NA                       NA
## 1.087671                  NA                NA                       NA
## 1.090411                  NA                NA                       NA
## 1.093151                  NA                NA                       NA
## 1.095890                  NA                NA                       NA
## 1.098630                  NA                NA                       NA
## 1.101370                  NA                NA                       NA
## 1.104110                  NA                NA                       NA
## 1.106849                  NA                NA                       NA
## 1.109589                  NA                NA                       NA
## 1.112329                  NA                NA                       NA
## 1.115068                  NA                NA                       NA
## 1.117808                  NA                NA                       NA
## 1.120548                  NA                NA                       NA
## 1.123288                  NA                NA                       NA
## 1.126027                  NA                NA                       NA
## 1.128767                  NA                NA                       NA
## 1.131507                  NA                NA                       NA
## 1.134247                  NA                NA                       NA
## 1.136986                  NA                NA                       NA
## 1.139726                  NA                NA                       NA
## 1.142466                  NA                NA                       NA
## 1.145205                  NA                NA                       NA
## 1.147945                  NA                NA                       NA
## 1.150685                  NA                NA                       NA
## 1.153425                  NA                NA                       NA
## 1.156164                  NA                NA                       NA
## 1.158904                  NA                NA                       NA
## 1.161644                  NA                NA                       NA
## 1.164384                  NA                NA                       NA
## 1.167123                  NA                NA                       NA
## 1.169863                  NA                NA                       NA
## 1.172603                  NA                NA                       NA
## 1.175342                  NA                NA                       NA
## 1.178082                  NA                NA                       NA
## 1.180822                  NA                NA                       NA
## 1.183562                  NA                NA                       NA
## 1.186301                  NA                NA                       NA
## 1.189041                  NA                NA                       NA
## 1.191781                  NA                NA                       NA
## 1.194521                  NA                NA                       NA
## 1.197260                  NA                NA                       NA
## 1.200000                  NA                NA                       NA
## 1.202740                  NA                NA                       NA
## 1.205479                  NA                NA                       NA
## 1.208219                  NA                NA                       NA
## 1.210959                  NA                NA                       NA
## 1.213699                  NA                NA                       NA
## 1.216438                  NA                NA                       NA
## 1.219178                  NA                NA                       NA
## 1.221918                  NA                NA                       NA
## 1.224658                  NA                NA                       NA
## 1.227397                  NA                NA                       NA
## 1.230137                  NA                NA                       NA
## 1.232877                  NA                NA                       NA
## 1.235616                  NA                NA                       NA
## 1.238356                  NA                NA                       NA
## 1.241096                  NA                NA                       NA
## 1.243836                  NA                NA                       NA
## 1.246575                  NA                NA                       NA
## 1.249315                  NA                NA                       NA
## 1.252055                  NA                NA                       NA
## 1.254795                  NA                NA                       NA
## 1.257534                  NA                NA                       NA
## 1.260274                  NA                NA                       NA
## 1.263014                  NA                NA                       NA
## 1.265753                  NA                NA                       NA
## 1.268493                  NA                NA                       NA
## 1.271233                  NA                NA                       NA
## 1.273973                  NA                NA                       NA
## 1.276712                  NA                NA                       NA
## 1.279452                  NA                NA                       NA
## 1.282192                  NA                NA                       NA
## 1.284932                  NA                NA                       NA
## 1.287671                  NA                NA                       NA
## 1.290411                  NA                NA                       NA
## 1.293151                  NA                NA                       NA
## 1.295890                  NA                NA                       NA
## 1.298630                  NA                NA                       NA
## 1.301370                  NA                NA                       NA
## 1.304110                  NA                NA                       NA
## 1.306849                  NA                NA                       NA
## 1.309589                  NA                NA                       NA
## 1.312329                  NA                NA                       NA
## 1.315068                  NA                NA                       NA
## 1.317808                  NA                NA                       NA
## 1.320548                  NA                NA                       NA
## 1.323288                  NA                NA                       NA
## 1.326027                  NA                NA                       NA
## 1.328767                  NA                NA                       NA
## 1.331507                  NA                NA                       NA
## 1.334247                  NA                NA                       NA
## 1.336986                  NA                NA                       NA
## 1.339726                  NA                NA                       NA
## 1.342466                  NA                NA                       NA
## 1.345205                  NA                NA                       NA
## 1.347945                  NA                NA                       NA
## 1.350685                  NA                NA                       NA
## 1.353425                  NA                NA                       NA
## 1.356164                  NA                NA                       NA
## 1.358904                  NA                NA                       NA
## 1.361644                  NA                NA                       NA
## 1.364384                  NA                NA                       NA
## 1.367123                  NA                NA                       NA
## 1.369863                  NA                NA                       NA
## 1.372603                  NA                NA                       NA
## 1.375342                  NA                NA                       NA
## 1.378082                  NA                NA                       NA
## 1.380822                  NA                NA                       NA
## 1.383562                  NA                NA                       NA
## 1.386301                  NA                NA                       NA
## 1.389041                  NA                NA                       NA
## 1.391781                  NA                NA                       NA
## 1.394521                  NA                NA                       NA
## 1.397260                  NA                NA                       NA
## 1.400000                  NA                NA                       NA
## 1.402740                  NA                NA                       NA
## 1.405479                  NA                NA                       NA
## 1.408219                  NA                NA                       NA
## 1.410959                  NA                NA                       NA
## 1.413699                  NA                NA                       NA
## 1.416438                  NA                NA                       NA
## 1.419178                  NA                NA                       NA
## 1.421918                  NA                NA                       NA
## 1.424658                  NA                NA                       NA
## 1.427397                  NA                NA                       NA
## 1.430137                  NA                NA                       NA
## 1.432877                  NA                NA                       NA
## 1.435616                  NA                NA                       NA
## 1.438356                  NA                NA                       NA
## 1.441096                  NA                NA                       NA
## 1.443836                  NA                NA                       NA
## 1.446575                  NA                NA                       NA
## 1.449315                  NA                NA                       NA
## 1.452055                  NA                NA                       NA
## 1.454795                  NA                NA                       NA
## 1.457534                  NA                NA                       NA
## 1.460274                  NA                NA                       NA
## 1.463014                  NA                NA                       NA
## 1.465753                  NA                NA                       NA
## 1.468493                  NA                NA                       NA
## 1.471233                  NA                NA                       NA
## 1.473973                  NA                NA                       NA
## 1.476712                  NA                NA                       NA
## 1.479452                  NA                NA                       NA
## 1.482192                  NA                NA                       NA
## 1.484932                  NA                NA                       NA
## 1.487671                  NA                NA                       NA
## 1.490411                  NA                NA                       NA
## 1.493151                  NA                NA                       NA
## 1.495890                  NA                NA                       NA
## 1.498630        -160.7065886        604.662288             -201.7833010
## 1.501370         -98.1592798        258.493234             -200.2853072
## 1.504110         115.1938966       -308.840516              153.6952664
## 1.506849         -35.2460536        -17.735959               13.0306587
## 1.509589         -87.2512931         48.148493               -0.8485534
## 1.512329         391.1530605       -830.690351              399.5859373
## 1.515068         226.9350124       -528.262474              261.3761083
## 1.517808         -17.5623732        -62.793730               40.4047500
## 1.520548        -291.0348664        464.008790             -212.9252774
## 1.523288         146.5583549       -388.287241              201.7775329
## 1.526027        -220.6812707        291.153654             -110.4237365
## 1.528767        -334.0285127        502.872795             -208.7956360
## 1.531507        -489.9150333        763.382823             -313.4191428
## 1.534247        -395.8713919        575.301019             -219.3809810
## 1.536986        -106.9544720         14.754851               52.2482677
## 1.539726        -487.8494309        745.643399             -297.7453213
## 1.542466        -721.3831296       1183.529974             -502.0981980
## 1.545205        -305.8279838        385.200505             -119.3238742
## 1.547945        -172.7926571        110.483276               22.3580278
## 1.550685        -295.6284770        384.757655             -129.0805318
## 1.553425        -446.5030224        630.512226             -223.9605567
## 1.556164        -315.1378834        386.165509             -110.9789793
## 1.558904        -151.4350432         60.420103               51.0635869
## 1.561644        -327.8223400        379.074148              -91.2031619
## 1.564384        -822.2969403       1288.168554             -505.8229677
## 1.567123        -770.2780229       1208.141679             -477.8150092
## 1.569863       -1151.5571437       1891.505400             -779.8996094
## 1.572603        -560.4421482        819.321984             -298.8311893
## 1.575342        -946.1113172       1537.030185             -630.8702213
## 1.578082        -806.3252416       1263.123787             -496.7498992
## 1.580822        -907.0857871       1433.332550             -566.1981159
## 1.583562        -826.8964014       1229.923641             -442.9785932
## 1.586301        -381.9663010        605.792207             -263.7772599
## 1.589041        2363.3104686      -4529.087359             2125.8255371
## 1.591781        2430.2910805      -4652.875980             2182.6335462
## 1.594521        2338.5493138      -4431.258200             2052.7575330
## 1.597260        2311.5746214      -4368.475939             2016.9499639
## 1.600000        2245.1922262      -4332.075532             2046.9319522
## 1.602740        2714.7634524      -5166.165930             2411.4511237
## 1.605479        2013.0233154      -3852.521272             1799.5466031
## 1.608219        2285.5496832      -4359.957569             2034.4565325
## 1.610959        2501.3378750      -4745.512035             2204.2228065
## 1.613699        2014.8037746      -3827.569861             1772.8147335
## 1.616438        2441.8720643      -4660.766715             2178.9432972
## 1.619178        2287.8391237      -4358.703573             2030.9130963
## 1.621918        2282.4476716      -4324.802861             2002.4038360
## 1.624658        2293.5818634      -4373.493162             2039.9599456
## 1.627397        2010.6227991      -3841.635308             1791.0611552
## 1.630137        2248.1234682      -4279.595550             1991.5207285
## 1.632877        2418.2587766      -4595.583975             2137.3738451
## 1.635616        2584.8465143      -4887.981368             2263.1835006
## 1.638356        2452.3613246      -4615.695920             2123.3832424
## 1.641096        2253.7413678      -4274.656007             1980.9632856
## 1.643836        2635.5995413      -4992.862765             2317.3118700
## 1.646575        2493.1530390      -4702.736883             2169.6324910
## 1.649315        2148.2520070      -4038.258107             1850.0547468
## 1.652055        2780.8307924      -5252.421157             2431.6390116
## 1.654795        2443.6445849      -4629.539153             2145.9432150
## 1.657534        2207.7881739      -4108.979756             1861.2402287
## 1.660274         447.0011693       -902.942733              415.9902104
## 1.663014        -142.2531229        198.445303              -96.1435339
## 1.665753         231.8365575       -498.846386              227.0584753
## 1.668493        -113.7479905        174.317230             -100.5205932
## 1.671233        -583.1296526       1115.307952             -572.1296526
## 1.673973        -610.7232989       1129.689765             -558.9178195
## 1.676712          76.8778377       -175.252234               58.4230432
## 1.679452        -136.7783383        245.114912             -148.2879274
## 1.682192        -288.1137904        500.791296             -252.6288589
## 1.684932         421.1087484       -863.637343              402.5772416
## 1.687671        -311.4958086        557.500538             -285.9560826
## 1.690411        -380.9512607        677.416921             -336.4170141
## 1.693151         113.4070540       -247.028475               93.6700677
## 1.695890        -566.0004529       1092.268730             -566.2196310
## 1.698630        -336.0074276        589.277200             -293.2211262
## 1.701370        -337.8071992        604.819209             -306.9633636
## 1.704110        -528.5825442        954.452091             -465.8209004
## 1.706849        -367.6848663        643.201941             -315.4684280
## 1.709589        -223.8189967        387.078421             -203.2107775
## 1.712329        -185.3828871        285.466475             -140.0349419
## 1.715068        -502.6171976        935.521398             -472.8555537
## 1.717808        -629.3956019       1193.048070             -603.6038211
## 1.720548           0.4617547        -18.946096              -21.4670124
## 1.723288        -130.8386563        253.750617             -162.8633138
## 1.726027        -345.7377497        654.674831             -348.8884346
## 1.728767        -411.7559257        791.042690             -419.2381175
## 1.731507        -961.9514873       1827.266690             -905.2665558
## 1.734247        -337.8726994        638.114593             -340.1932474
## 1.736986        -564.4902910       1100.440187             -575.9012499
## 1.739726        -620.9527866       1163.419973             -582.4185400
## 1.742466        -209.9473412        381.151548             -211.1555604
## 1.745205        -663.2267434       1255.184325             -631.9089352
## 1.747945        -394.5223690        733.882426             -379.3114101
## 1.750685       -1141.6989759       2210.876735            -1109.1291129
## 1.753425        -181.7598460        270.713544             -128.9050515
## 1.756164        -336.2010606        583.590494             -287.3407866
## 1.758904         -96.1930697        105.678622              -49.4369053
## 1.761644        -449.2136201        818.799175             -409.5369078
## 1.764384        -324.8117364        527.209106             -242.3487227
## 1.767123        -265.9711154        416.958001             -190.9382387
## 1.769863        -315.5586131        509.431626             -233.8243665
## 1.772603        -400.0877090        649.478859             -289.3425035
## 1.775342        -114.4128323         72.208558                2.2529211
## 1.778082        -359.9774417        616.745996             -296.7199074
## 1.780822        -136.7981590        147.749074              -50.9022685
## 1.783562         -65.1526795         11.430718               13.7706082
## 1.786301        -293.8925060        449.513111             -195.5719580
## 1.789041        -202.2204487        277.695023             -115.4259282
## 1.791781           4.6449643       -131.624844               87.0285260
## 1.794521        -276.9210033        386.471475             -149.5018252
## 1.797260        -522.2540286        877.143005             -394.8403300
## 1.800000        -181.0365126        205.568247              -64.4830880
## 1.802740        -137.3386314        140.177964              -42.7906862
## 1.805479        -209.9665126        263.428247              -93.4130880
## 1.808219          36.2658369       -214.493986              138.2767958
## 1.810959        -317.5510672        453.506945             -175.9072316
## 1.813699        -412.9675969        634.361923             -261.3456791
## 1.816438        -519.3931860        816.087073             -336.6452408
## 1.819178        -371.3100079        542.465923             -211.1072681
## 1.821918        -357.3400079        522.525923             -205.1372681
## 1.824658        -507.4788572        803.548827             -336.0213229
## 1.827397        -501.8217248        782.774288             -320.9039165
## 1.830137        -458.2561332        684.459543             -266.1547634
## 1.832877        -213.2832980        251.576886              -78.2449419
## 1.835616        -979.9726770       1648.254275             -708.2329510
## 1.838356        -334.9099647        553.501453             -258.5428414
## 1.841096         -60.2362910         64.362324              -44.0773869
## 1.843836        -435.1961631        770.838233             -375.5934234
## 1.846575        -362.9443524        606.635982             -283.6429826
## 1.849315          81.5784487       -198.661676               77.1318734
## 1.852055        -323.6782362        561.131146             -277.4042636
## 1.854795         -63.4745923         60.627968              -37.1047293
## 1.857534        -531.5162179        942.111219             -450.5463549
## 1.860274         362.8363325       -730.810320              328.0226339
## 1.863014        -384.2975878        660.734233             -316.3879987
## 1.865753        -128.0892931        211.268329             -123.1303890
## 1.868493        -405.9864163        761.350246             -395.3151835
## 1.871233        -419.9243524        781.563105             -401.5901058
## 1.873973        -454.7450830        848.626484             -433.8327542
## 1.876712        -256.0091013        468.258630             -252.2008821
## 1.879452        -121.8491768        166.133302              -84.2354782
## 1.882192        -675.9358435       1276.306635             -640.3221449
## 1.884932        -106.5148688        177.820850             -111.2573346
## 1.887671         -12.0128414         44.238713              -72.1772250
## 1.890411        -267.0218485        504.697823             -277.6273279
## 1.893151        -519.6665151        995.647430             -515.9322685
## 1.895890         195.4134616       -378.372797              143.0079822
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## 1.901370        -358.6742602        687.501277             -368.7783698
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## 1.906849        -194.7889244        332.552523             -177.7149518
## 1.909589          92.6901690       -162.405664               29.7641416
## 1.912329        -590.1318908       1159.443935             -609.2633976
## 1.915068        -752.7572606       1460.960428             -748.1545209
## 1.917808        -500.2497471        961.304305             -501.0059115
## 1.920548         -98.2719347        252.754160             -194.4335786
## 1.923288        -310.5843914        545.836607             -275.2035694
## 1.926027       -1018.8610306       1869.598105             -890.6884279
## 1.928767        -647.0357272       1198.533800             -591.4494258
## 1.931507       -1080.4631153       2038.868028             -998.3562660
## 1.934247        -878.0584345       1661.420310             -823.3132291
## 1.936986       -1038.9744711       1916.827726             -917.8046081
## 1.939726        -925.2799115       1680.016689             -794.6881307
## 1.942466        -967.7830348       1817.116086             -889.2844047
## 1.945205        -796.9804437       1465.762959             -708.7338683
## 1.947945       -1253.2338799       2314.730105            -1101.4475786
## 1.950685       -1248.8264802       2345.200237            -1136.3251103
## 1.953425       -1156.7768845       2143.218854            -1026.3933229
## 1.956164        -992.0259688       1812.569077             -860.4944619
## 1.958904       -1288.2767152       2380.311666            -1131.9863042
## 1.961644       -1386.6658177       2596.196720            -1249.4822561
## 1.964384       -1205.7503930       2220.648063            -1054.8490232
## 1.967123        -962.7190738       1791.911452             -869.1437313
## 1.969863        -929.2646786       1724.183483             -834.8701581
## 1.972603        -769.2268978       1412.006552             -682.7310074
## 1.975342       -1216.4995412       2237.154578            -1060.6063905
## 1.978082        -757.9039547       1363.793542             -645.8409410
## 1.980822        -608.4827401       1097.956592             -529.4252058
## 1.983562        -928.7185599       1737.688506             -848.9212996
## 1.986301       -1390.3123175       2613.725336            -1263.3643723
## 1.989041        -644.0924063       1209.611541             -605.4704885
## 1.991781       -1249.4742195       2307.375168            -1097.8523017
## 1.994521       -1003.9414889       1809.624775             -845.6346396
## 1.997260        -882.1852033       1627.534122             -785.3002718
## 2.000000        -286.9863748        505.640574             -258.6055529
## 2.002740       -1044.9239912       1926.578821             -921.6061830
## 2.005479        -334.8216483        589.141258             -294.2709634
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## 2.013699        -114.5016251        183.092993             -108.5427210
## 2.016438        -906.2141457       1645.668719             -779.4059265
## 2.019178         104.1371511       -172.094149               28.0056443
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## 2.024658        -526.5592299       1037.019161             -550.4112847
## 2.027397          91.8330557       -195.211985               63.4275762
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## 2.032877         637.9615281      -1251.444273              573.5313911
## 2.035616        -149.1544421        188.957531              -79.7544421
## 2.038356         256.3734733       -557.111999              260.7871719
## 2.041096         343.0058934       -655.916565              272.9593180
## 2.043836         211.3798776       -391.820698              140.4894666
## 2.046575       -1069.9217920       1997.081272             -967.1108331
## 2.049315         381.4790183       -627.372404              205.9420320
## 2.052055         107.3450822       -254.537408              107.2409727
## 2.054795         684.2863267      -1222.441815              498.2041350
## 2.057534        -162.4766230         70.558057               51.9672126
## 2.060274        -709.2311685       1082.072627             -412.7928123
## 2.063014        -335.6176824        643.612778             -347.9464495
## 2.065753        -991.1810132       1591.750399             -640.5207393
## 2.068493         129.4425733       -202.559788               33.1658610
## 2.071233          89.5247012       -181.507605               52.0315505
## 2.073973        -941.2128646       1734.181225             -832.9197140
## 2.076712        -416.1724836        608.240189             -232.0190589
## 2.079452         164.4979158       -212.281432                7.8321624
## 2.082192         153.4777174       -315.284870              121.8557995
## 2.084932         368.9902719       -655.978473              247.0368473
## 2.087671        -348.0242420        752.825898             -444.7530091
## 2.090411          32.7553313       -177.009961              104.3032765
## 2.093151         468.8374633       -972.420801              463.6319839
## 2.095890         -76.8147119        132.469851              -95.6064927
## 2.098630         195.3397472       -316.351396               81.0602952
## 2.101370        -208.4084520        367.849112             -199.3920136
## 2.104110         345.5514019       -766.199363              380.6966074
## 2.106849         520.0222238       -921.573883              361.6003060
## 2.109589         189.8932632       -420.715962              190.8713454
## 2.112329         -46.4782627         54.755857              -48.2289476
## 2.115068          77.6852533       -195.916381               78.2797738
## 2.117808         309.2993604       -627.308978              278.0582645
## 2.120548        -225.7017081        379.487679             -193.7373245
## 2.123288         300.4507510       -492.962444              152.5603400
## 2.126027         164.1428971       -314.026189              109.9319382
## 2.128767        -300.2689501        553.271478             -292.9538816
## 2.131507        -935.2895437       1731.098967             -835.7607766
## 2.134247         344.0317556       -624.431303              240.4481939
## 2.136986         -53.4235138        -30.027613               43.4997738
## 2.139726         546.2506937      -1027.995207              441.7931595
## 2.142466         387.3222578       -684.839705              257.5660935
## 2.145205        -770.1587401       1470.637360             -740.4299729
## 2.147945        -426.6282195        554.510565             -167.8336990
## 2.150685        -210.6712764        326.287090             -155.5671668
## 2.153425         984.7331935      -1808.324590              783.6400428
## 2.156164         197.4895704       -384.234604              146.7936800
## 2.158904         322.7264837       -507.839937              145.1621001
## 2.161644        -565.7853337       1102.268629             -576.4346487
## 2.164384        -315.4718634        606.192373             -330.6718634
## 2.167123         411.3331802       -810.228673              358.9441391
## 2.169863        -312.2896326        568.767638             -296.4293586
## 2.172603         102.7591096       -301.124367              158.4139042
## 2.175342        -154.5223125        213.068614              -98.4976550
## 2.178082         -18.5882420         30.384035              -51.7471461
## 2.180822         820.3528631      -1613.813244              753.5090275
## 2.183562         -10.7118700         -8.582408              -20.6570755
## 2.186301         535.3729951      -1127.617892              552.2935430
## 2.189041         155.2761159       -326.067969              130.8404995
## 2.191781        -185.7233677        295.125519             -149.3535047
## 2.194521        -253.6864321        324.059867             -110.3247882
## 2.197260         243.6742139       -512.099781              228.4742139
## 2.200000         695.1703600      -1361.503032              626.3813189
## 2.202740         483.5787826       -963.925357              440.3952209
## 2.205479       -1267.3040186       2365.075862            -1137.7231967
## 2.208219       -1038.2479206       1894.596543             -896.2999754
## 2.210959        -528.5533021        999.119634             -510.5176857
## 2.213699          29.9759764       -230.010156              160.0828257
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## 2.221918         -70.8562129        137.374771             -106.4699115
## 2.224658         191.5849993       -407.513133              175.9767801
## 2.227397        -773.1979231       1433.956822             -700.7102519
## 2.230137         390.2778394       -697.076895              266.8477024
## 2.232877         356.6187253       -677.961407              281.3913280
## 2.235616         541.3677506      -1098.333430              517.0143259
## 2.238356           2.9773147        -15.585435              -27.3432332
## 2.241096        -555.1089094        792.187013             -277.0294574
## 2.243836        -361.8594615        512.573049             -190.6649410
## 2.246575         156.7906066       -381.491471              184.7495107
## 2.249315         561.3514193       -980.777480              379.4747070
## 2.252055         529.0671205       -955.252718              386.2342437
## 2.254795         684.0202869      -1343.216585              619.2449445
## 2.257534         311.3767917       -655.003567              303.6754218
## 2.260274         282.2763143       -529.175215              206.9475472
## 2.263014        -666.8936309       1253.024949             -626.0826720
## 2.265753        -981.3429941       1808.688059             -867.2964188
## 2.268493        -163.1471676        231.052571             -107.8567567
## 2.271233         242.0540470       -441.344379              159.3389785
## 2.273973         393.0072433       -790.272689              357.3140927
## 2.276712         -68.0627567          4.867311               23.2440927
## 2.279452         -33.9008729        -34.631799               28.5813189
## 2.282192         837.2494376      -1642.485845              765.2850540
## 2.284932        -688.9133752       1220.730191             -571.7681697
## 2.287671       -1185.2844619       2180.360036            -1035.0269277
## 2.290411         153.0505268       -405.066106              212.0642255
## 2.293151         538.8485518      -1036.859416              458.0595107
## 2.295890         252.4711894       -469.482773              177.0602305
## 2.298630         692.1905360      -1291.954343              559.8124538
## 2.301370         240.2724995       -446.003202              165.7793488
## 2.304110         270.1509013       -517.458635              207.3563807
## 2.306849        -657.7673362       1167.092908             -549.2769252
## 2.309589        -173.7200028        186.244817              -52.4761672
## 2.312329         294.8902346       -593.468809              258.6272209
## 2.315068         468.0331205       -905.584718              397.6002437
## 2.317808        -230.6063316        278.913365              -88.2583864
## 2.320548         161.0826090       -299.015202               97.9812392
## 2.323288         322.6724173       -532.096188              169.4724173
## 2.326027        -365.4214732        526.110771             -200.6406512
## 2.328767         -21.7728430        141.284744             -159.4632540
## 2.331507         645.0593720      -1277.253659              592.2429336
## 2.334247        -258.4381465        442.409871             -223.9230780
## 2.336986        -193.3353976        316.034510             -162.6504661
## 2.339726         120.8603899       -248.488572               87.6768282
## 2.342466        -884.1133727       1650.165803             -806.0037837
## 2.345205         345.1020794       -687.270581              302.2171479
## 2.347945        -945.7296533       1729.149049             -823.3707492
## 2.350685        -451.8091693        873.371095             -461.5132789
## 2.353425         532.0363467       -956.517197              384.5294974
## 2.356164         215.0660636       -453.001289              197.9838718
## 2.358904        -136.8797355        129.925926              -32.9975437
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## 2.372603         -91.0873769        146.985044              -95.8490207
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## 2.380822         461.5558195       -828.035595              326.5284222
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## 2.389041        -102.2926604        189.351776             -127.0104686
## 2.391781        -138.8762951        182.220415              -83.2954731
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## 2.400000        -277.7571261        449.075228             -211.2694549
## 2.402740        1010.2326340      -1867.120731              816.9367435
## 2.405479        -229.7926902        266.962794              -77.1214574
## 2.408219         336.9654193       -544.936987              168.0202139
## 2.410959         615.4206298      -1210.817271              555.4452873
## 2.413699          52.5157755        -77.339069              -15.1280601
## 2.416438         237.5480154       -476.773412              199.2740428
## 2.419178         365.3760819       -785.418586              380.0911503
## 2.421918         262.9761225       -588.692640              285.7651636
## 2.424658         605.9091454      -1210.840877              564.9803782
## 2.427397        -148.2815919        190.904981              -82.5747426
## 2.430137        -331.2981813        584.499804             -293.1529758
## 2.432877         233.9405584       -524.454388              250.5624762
## 2.435616        -197.8609576        287.962343             -130.0527384
## 2.438356        -336.2655811        562.281179             -265.9669509
## 2.441096         302.7208780       -606.069822              263.3975904
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## 2.446575        -103.3206330        144.109091              -80.7398110
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## 2.460274         218.9896501       -428.040243              169.0992392
## 2.463014        -355.2569252        654.094004             -338.7884321
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## 2.468493         376.7723301       -765.997384              349.2737000
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## 2.493151        -470.5864819        915.569556             -484.9344271
## 2.495890         174.5668515       -410.737111              196.2189062
## 2.498630        -160.7065886       -130.555079              -30.8134379
## 2.501370                  NA                NA                       NA
## 2.504110                  NA                NA                       NA
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## 2.912329                  NA                NA                       NA
## 2.915068                  NA                NA                       NA
## 2.917808                  NA                NA                       NA
## 2.920548                  NA                NA                       NA
## 2.923288                  NA                NA                       NA
## 2.926027                  NA                NA                       NA
## 2.928767                  NA                NA                       NA
## 2.931507                  NA                NA                       NA
## 2.934247                  NA                NA                       NA
## 2.936986                  NA                NA                       NA
## 2.939726                  NA                NA                       NA
## 2.942466                  NA                NA                       NA
## 2.945205                  NA                NA                       NA
## 2.947945                  NA                NA                       NA
## 2.950685                  NA                NA                       NA
## 2.953425                  NA                NA                       NA
## 2.956164                  NA                NA                       NA
## 2.958904                  NA                NA                       NA
## 2.961644                  NA                NA                       NA
## 2.964384                  NA                NA                       NA
## 2.967123                  NA                NA                       NA
## 2.969863                  NA                NA                       NA
## 2.972603                  NA                NA                       NA
## 2.975342                  NA                NA                       NA
## 2.978082                  NA                NA                       NA
## 2.980822                  NA                NA                       NA
## 2.983562                  NA                NA                       NA
## 2.986301                  NA                NA                       NA
## 2.989041                  NA                NA                       NA
## 2.991781                  NA                NA                       NA
## 2.994521                  NA                NA                       NA
## 2.997260                  NA                NA                       NA
## 
## $figure
##   [1]   286.9863748  1044.9239912   334.8216483  -561.4433654  -127.3880984
##   [6]   114.5016251   906.2141457  -104.1371511   -53.7951810   526.5592299
##  [11]   -91.8330557  -564.4173795  -637.9615281   149.1544421  -256.3734733
##  [16]  -343.0058934  -211.3798776  1069.9217920  -381.4790183  -107.3450822
##  [21]  -684.2863267   162.4766230   709.2311685   335.6176824   991.1810132
##  [26]  -129.4425733   -89.5247012   941.2128646   416.1724836  -164.4979158
##  [31]  -153.4777174  -368.9902719   348.0242420   -32.7553313  -468.8374633
##  [36]    76.8147119  -195.3397472   208.4084520  -345.5514019  -520.0222238
##  [41]  -189.8932632    46.4782627   -77.6852533  -309.2993604   225.7017081
##  [46]  -300.4507510  -164.1428971   300.2689501   935.2895437  -344.0317556
##  [51]    53.4235138  -546.2506937  -387.3222578   770.1587401   426.6282195
##  [56]   210.6712764  -984.7331935  -197.4895704  -322.7264837   565.7853337
##  [61]   315.4718634  -411.3331802   312.2896326  -102.7591096   154.5223125
##  [66]    18.5882420  -820.3528631    10.7118700  -535.3729951  -155.2761159
##  [71]   185.7233677   253.6864321  -243.6742139  -695.1703600  -483.5787826
##  [76]  1267.3040186  1038.2479206   528.5533021   -29.9759764  -281.0326730
##  [81]  -436.6571338    70.8562129  -191.5849993   773.1979231  -390.2778394
##  [86]  -356.6187253  -541.3677506    -2.9773147   555.1089094   361.8594615
##  [91]  -156.7906066  -561.3514193  -529.0671205  -684.0202869  -311.3767917
##  [96]  -282.2763143   666.8936309   981.3429941   163.1471676  -242.0540470
## [101]  -393.0072433    68.0627567    33.9008729  -837.2494376   688.9133752
## [106]  1185.2844619  -153.0505268  -538.8485518  -252.4711894  -692.1905360
## [111]  -240.2724995  -270.1509013   657.7673362   173.7200028  -294.8902346
## [116]  -468.0331205   230.6063316  -161.0826090  -322.6724173   365.4214732
## [121]    21.7728430  -645.0593720   258.4381465   193.3353976  -120.8603899
## [126]   884.1133727  -345.1020794   945.7296533   451.8091693  -532.0363467
## [131]  -215.0660636   136.8797355  -596.6610134  -687.7398353  -484.7143832
## [136]   -64.9531686    91.0873769  -316.9195820  -343.7177556  -461.5558195
## [141]  -245.5614974  -268.0170839   102.2926604   138.8762951  -533.1365360
## [146]  1316.6171261   277.7571261 -1010.2326340   229.7926902  -336.9654193
## [151]  -615.4206298   -52.5157755  -237.5480154  -365.3760819  -262.9761225
## [156]  -605.9091454   148.2815919   331.2981813  -233.9405584   197.8609576
## [161]   336.2655811  -302.7208780   381.3827359   103.3206330   -23.8432616
## [166]  -576.7468623  -527.8658303  -491.2574650  -218.9896501   355.2569252
## [171]  -526.5304380  -376.7723301  -299.0235447   788.3041722  -783.1729561
## [176]  -373.6257871   212.1452083  -424.5300403   503.6110439   184.3641166
## [181]   470.5864819  -174.5668515   160.7065886    98.1592798  -115.1938966
## [186]    35.2460536    87.2512931  -391.1530605  -226.9350124    17.5623732
## [191]   291.0348664  -146.5583549   220.6812707   334.0285127   489.9150333
## [196]   395.8713919   106.9544720   487.8494309   721.3831296   305.8279838
## [201]   172.7926571   295.6284770   446.5030224   315.1378834   151.4350432
## [206]   327.8223400   822.2969403   770.2780229  1151.5571437   560.4421482
## [211]   946.1113172   806.3252416   907.0857871   826.8964014   381.9663010
## [216] -2363.3104686 -2430.2910805 -2338.5493138 -2311.5746214 -2245.1922262
## [221] -2714.7634524 -2013.0233154 -2285.5496832 -2501.3378750 -2014.8037746
## [226] -2441.8720643 -2287.8391237 -2282.4476716 -2293.5818634 -2010.6227991
## [231] -2248.1234682 -2418.2587766 -2584.8465143 -2452.3613246 -2253.7413678
## [236] -2635.5995413 -2493.1530390 -2148.2520070 -2780.8307924 -2443.6445849
## [241] -2207.7881739  -447.0011693   142.2531229  -231.8365575   113.7479905
## [246]   583.1296526   610.7232989   -76.8778377   136.7783383   288.1137904
## [251]  -421.1087484   311.4958086   380.9512607  -113.4070540   566.0004529
## [256]   336.0074276   337.8071992   528.5825442   367.6848663   223.8189967
## [261]   185.3828871   502.6171976   629.3956019    -0.4617547   130.8386563
## [266]   345.7377497   411.7559257   961.9514873   337.8726994   564.4902910
## [271]   620.9527866   209.9473412   663.2267434   394.5223690  1141.6989759
## [276]   181.7598460   336.2010606    96.1930697   449.2136201   324.8117364
## [281]   265.9711154   315.5586131   400.0877090   114.4128323   359.9774417
## [286]   136.7981590    65.1526795   293.8925060   202.2204487    -4.6449643
## [291]   276.9210033   522.2540286   181.0365126   137.3386314   209.9665126
## [296]   -36.2658369   317.5510672   412.9675969   519.3931860   371.3100079
## [301]   357.3400079   507.4788572   501.8217248   458.2561332   213.2832980
## [306]   979.9726770   334.9099647    60.2362910   435.1961631   362.9443524
## [311]   -81.5784487   323.6782362    63.4745923   531.5162179  -362.8363325
## [316]   384.2975878   128.0892931   405.9864163   419.9243524   454.7450830
## [321]   256.0091013   121.8491768   675.9358435   106.5148688    12.0128414
## [326]   267.0218485   519.6665151  -195.4134616   141.8970431   358.6742602
## [331]  -176.4157398   194.7889244   -92.6901690   590.1318908   752.7572606
## [336]   500.2497471    98.2719347   310.5843914  1018.8610306   647.0357272
## [341]  1080.4631153   878.0584345  1038.9744711   925.2799115   967.7830348
## [346]   796.9804437  1253.2338799  1248.8264802  1156.7768845   992.0259688
## [351]  1288.2767152  1386.6658177  1205.7503930   962.7190738   929.2646786
## [356]   769.2268978  1216.4995412   757.9039547   608.4827401   928.7185599
## [361]  1390.3123175   644.0924063  1249.4742195  1003.9414889   882.1852033
## 
## $type
## [1] "additive"
## 
## attr(,"class")
## [1] "decomposed.ts"

plot(dpd_ts)

library(zoo)

rollmean(dpd_ts, 3)

## Time Series:
## Start = c(1, 2) 
## End = c(2, 364) 
## Frequency = 365 
##           date        ca nb_ventes
## 1.002740 18688 15757.938  938.0000
## 1.005479 18689 15297.646  918.0000
## 1.008219 18690 15955.377  919.0000
## 1.010959 18691 16150.907  935.6667
## 1.013699 18692 16009.526  936.6667
## 1.016438 18693 15412.246  931.6667
## 1.019178 18694 15387.322  928.6667
## 1.021918 18695 15628.970  940.6667
## 1.024658 18696 15336.173  936.0000
## 1.027397 18697 14915.530  914.6667
## 1.030137 18698 14524.853  895.3333
## 1.032877 18699 14668.397  895.3333
## 1.035616 18700 15232.757  905.3333
## 1.038356 18701 15781.723  925.6667
## 1.041096 18702 16046.290  940.3333
## 1.043836 18703 15686.000  941.6667
## 1.046575 18704 15544.010  932.3333
## 1.049315 18705 15486.850  928.0000
## 1.052055 18706 16286.473  949.0000
## 1.054795 18707 16323.567  949.6667
## 1.057534 18708 16385.662  941.6667
## 1.060274 18709 15599.676  906.6667
## 1.063014 18710 15434.719  909.3333
## 1.065753 18711 15480.053  908.6667
## 1.068493 18712 15525.517  911.6667
## 1.071233 18713 15838.348  908.0000
## 1.073973 18714 15266.420  896.0000
## 1.076712 18715 15129.469  899.0000
## 1.079452 18716 14939.707  900.0000
## 1.082192 18717 15293.136  911.0000
## 1.084932 18718 15678.316  932.0000
## 1.087671 18719 15673.198  941.6667
## 1.090411 18720 15871.465  931.6667
## 1.093151 18721 16368.116  939.0000
## 1.095890 18722 16800.019  943.0000
## 1.098630 18723 16597.232  958.3333
## 1.101370 18724 15773.795  936.6667
## 1.104110 18725 15419.612  925.0000
## 1.106849 18726 15337.482  926.6667
## 1.109589 18727 15813.179  950.3333
## 1.112329 18728 15869.076  964.6667
## 1.115068 18729 15980.320  957.6667
## 1.117808 18730 15822.640  940.0000
## 1.120548 18731 15701.253  935.3333
## 1.123288 18732 15427.560  935.3333
## 1.126027 18733 15387.519  937.0000
## 1.128767 18734 15188.005  934.3333
## 1.131507 18735 15434.765  938.3333
## 1.134247 18736 15355.199  936.0000
## 1.136986 18737 15505.153  928.0000
## 1.139726 18738 16015.019  953.0000
## 1.142466 18739 15896.482  941.6667
## 1.145205 18740 16286.039  957.0000
## 1.147945 18741 16421.196  960.6667
## 1.150685 18742 16442.582  972.6667
## 1.153425 18743 16254.382  971.3333
## 1.156164 18744 15519.386  959.3333
## 1.158904 18745 16030.096  978.3333
## 1.161644 18746 16061.090  985.3333
## 1.164384 18747 16234.336  962.3333
## 1.167123 18748 15767.586  928.6667
## 1.169863 18749 15544.706  915.0000
## 1.172603 18750 15873.427  918.6667
## 1.175342 18751 15963.767  923.3333
## 1.178082 18752 15568.729  905.3333
## 1.180822 18753 16107.482  913.0000
## 1.183562 18754 17103.126  945.0000
## 1.186301 18755 17758.983  959.3333
## 1.189041 18756 16967.877  944.0000
## 1.191781 18757 16196.059  919.0000
## 1.194521 18758 16200.758  923.3333
## 1.197260 18759 15965.558  928.0000
## 1.200000 18760 15434.636  914.3333
## 1.202740 18761 15074.960  895.6667
## 1.205479 18762 15101.180  885.0000
## 1.208219 18763 15416.193  883.6667
## 1.210959 18764 15490.733  878.6667
## 1.213699 18765 15560.783  890.6667
## 1.216438 18766 15503.959  892.0000
## 1.219178 18767 15473.172  896.3333
## 1.221918 18768 15523.781  890.3333
## 1.224658 18769 15535.742  898.3333
## 1.227397 18770 15560.949  901.0000
## 1.230137 18771 15523.090  895.6667
## 1.232877 18772 15591.483  892.6667
## 1.235616 18773 15881.190  896.6667
## 1.238356 18774 16378.787  925.0000
## 1.241096 18775 16367.710  943.6667
## 1.243836 18776 16505.160  957.0000
## 1.246575 18777 16089.272  937.6667
## 1.249315 18778 16298.919  925.6667
## 1.252055 18779 15720.869  908.6667
## 1.254795 18780 15841.177  910.6667
## 1.257534 18781 15236.897  888.6667
## 1.260274 18782 15797.242  897.0000
## 1.263014 18783 16199.109  897.6667
## 1.265753 18784 16800.439  914.3333
## 1.268493 18785 16735.950  900.3333
## 1.271233 18786 16081.779  893.0000
## 1.273973 18787 15750.182  891.6667
## 1.276712 18788 15619.559  894.6667
## 1.279452 18789 15939.300  895.3333
## 1.282192 18790 15840.003  880.0000
## 1.284932 18791 16020.357  889.0000
## 1.287671 18792 15645.600  879.3333
## 1.290411 18793 15881.806  891.0000
## 1.293151 18794 15673.817  879.0000
## 1.295890 18795 16151.521  882.0000
## 1.298630 18796 16202.048  879.0000
## 1.301370 18797 16280.167  884.3333
## 1.304110 18798 15910.229  881.3333
## 1.306849 18799 15994.942  902.0000
## 1.309589 18800 16400.309  907.0000
## 1.312329 18801 17230.096  928.0000
## 1.315068 18802 17315.686  912.6667
## 1.317808 18803 16961.446  905.6667
## 1.320548 18804 16338.367  893.0000
## 1.323288 18805 16011.363  900.6667
## 1.326027 18806 16066.827  897.6667
## 1.328767 18807 16053.040  892.0000
## 1.331507 18808 16413.617  877.3333
## 1.334247 18809 16780.637  865.0000
## 1.336986 18810 16557.972  853.3333
## 1.339726 18811 16330.629  836.0000
## 1.342466 18812 15530.592  820.0000
## 1.345205 18813 15435.453  806.0000
## 1.347945 18814 15681.683  810.3333
## 1.350685 18815 16218.667  823.6667
## 1.353425 18816 16139.447  819.3333
## 1.356164 18817 15405.529  795.6667
## 1.358904 18818 14499.616  788.6667
## 1.361644 18819 15200.812  797.0000
## 1.364384 18820 15723.086  810.3333
## 1.367123 18821 16134.742  803.0000
## 1.369863 18822 15584.059  796.0000
## 1.372603 18823 15607.442  811.3333
## 1.375342 18824 15967.242  817.3333
## 1.378082 18825 15961.433  825.3333
## 1.380822 18826 15776.280  799.3333
## 1.383562 18827 15529.263  790.3333
## 1.386301 18828 15249.057  773.3333
## 1.389041 18829 14860.853  769.3333
## 1.391781 18830 14627.523  768.6667
## 1.394521 18831 14357.357  756.3333
## 1.397260 18832 14714.030  752.0000
## 1.400000 18833 14350.007  743.0000
## 1.402740 18834 15102.783  768.3333
## 1.405479 18835 15248.670  778.6667
## 1.408219 18836 16272.890  798.3333
## 1.410959 18837 16241.157  792.6667
## 1.413699 18838 16089.417  792.3333
## 1.416438 18839 15499.590  792.6667
## 1.419178 18840 15317.440  799.3333
## 1.421918 18841 15410.817  816.6667
## 1.424658 18842 15412.243  811.3333
## 1.427397 18843 15319.063  813.3333
## 1.430137 18844 15487.092  813.3333
## 1.432877 18845 15378.745  815.0000
## 1.435616 18846 15517.898  813.3333
## 1.438356 18847 15879.173  827.0000
## 1.441096 18848 15981.713  837.0000
## 1.443836 18849 15617.640  835.0000
## 1.446575 18850 14845.577  809.0000
## 1.449315 18851 14674.200  801.0000
## 1.452055 18852 15092.489  816.3333
## 1.454795 18853 15417.752  819.0000
## 1.457534 18854 15483.247  833.3333
## 1.460274 18855 15874.118  836.6667
## 1.463014 18856 15552.866  845.0000
## 1.465753 18857 15836.756  843.0000
## 1.468493 18858 15231.666  832.3333
## 1.471233 18859 15235.052  839.0000
## 1.473973 18860 15222.389  831.3333
## 1.476712 18861 15314.838  835.3333
## 1.479452 18862 15464.508  830.0000
## 1.482192 18863 15470.218  823.3333
## 1.484932 18864 15695.819  827.6667
## 1.487671 18865 16248.042  848.0000
## 1.490411 18866 16321.817  849.3333
## 1.493151 18867 15918.733  835.0000
## 1.495890 18868 15809.600  832.0000
## 1.498630 18869 15965.547  836.6667
## 1.501370 18870 16217.885  889.3333
## 1.504110 18871 15974.532  919.3333
## 1.506849 18872 15905.767  982.3333
## 1.509589 18873 15639.939  972.3333
## 1.512329 18874 15381.372  967.6667
## 1.515068 18875 15324.376  958.3333
## 1.517808 18876 15988.308  981.6667
## 1.520548 18877 16066.382  988.6667
## 1.523288 18878 16255.126 1006.0000
## 1.526027 18879 16286.259 1021.6667
## 1.528767 18880 16886.018 1062.0000
## 1.531507 18881 17040.755 1084.0000
## 1.534247 18882 16804.149 1095.3333
## 1.536986 18883 16798.483 1100.0000
## 1.539726 18884 17109.977 1114.3333
## 1.542466 18885 17298.423 1123.3333
## 1.545205 18886 16984.889 1125.0000
## 1.547945 18887 16580.299 1107.3333
## 1.550685 18888 16712.918 1119.3333
## 1.553425 18889 16849.479 1122.0000
## 1.556164 18890 16691.546 1134.0000
## 1.558904 18891 16568.207 1139.0000
## 1.561644 18892 17042.749 1177.0000
## 1.564384 18893 17639.026 1207.6667
## 1.567123 18894 18423.102 1253.0000
## 1.569863 18895 18183.612 1235.0000
## 1.572603 18896 18355.142 1242.6667
## 1.575342 18897 18036.686 1222.3333
## 1.578082 18898 18363.250 1249.0000
## 1.580822 18899 18225.123 1272.0000
## 1.583562 18900 17865.687 1208.0000
## 1.586301 18901 14789.493 1015.0000
## 1.589041 18902 11744.507  804.3333
## 1.591781 18903  9159.438  669.6667
## 1.594521 18904  9228.882  650.6667
## 1.597260 18905  9394.762  667.0000
## 1.600000 18906  9024.077  661.0000
## 1.602740 18907  9296.373  688.0000
## 1.605479 18908  9274.597  670.3333
## 1.608219 18909  9486.120  672.3333
## 1.610959 18910  9493.857  662.6667
## 1.613699 18911  9342.453  658.6667
## 1.616438 18912  9542.977  672.0000
## 1.619178 18913  9289.383  659.3333
## 1.621918 18914  9435.852  662.3333
## 1.624658 18915  9703.119  674.6667
## 1.627397 18916  9732.519  682.3333
## 1.630137 18917  9623.573  673.3333
## 1.632877 18918  9088.955  639.3333
## 1.635616 18919  8913.934  615.3333
## 1.638356 18920  9076.624  618.0000
## 1.641096 18921  9024.189  619.0000
## 1.643836 18922  8983.430  620.6667
## 1.646575 18923  9100.563  612.3333
## 1.649315 18924  8969.957  602.3333
## 1.652055 18925  9010.582  611.0000
## 1.654795 18926  8966.029  594.6667
## 1.657534 18927 11193.489  700.3333
## 1.660274 18928 13667.193  814.6667
## 1.663014 18929 15532.943  928.3333
## 1.665753 18930 16081.843  943.0000
## 1.668493 18931 16537.879  931.3333
## 1.671233 18932 17364.435  950.3333
## 1.673973 18933 17187.775  940.0000
## 1.676712 18934 16751.322  932.6667
## 1.679452 18935 16435.773  927.3333
## 1.682192 18936 16091.380  927.3333
## 1.684932 18937 16249.273  939.6667
## 1.687671 18938 16334.620  942.6667
## 1.690411 18939 16640.863  942.3333
## 1.693151 18940 16906.260  934.0000
## 1.695890 18941 16863.932  933.6667
## 1.698630 18942 17298.632  950.6667
## 1.701370 18943 17240.155  971.6667
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## 2.912329 19385 16521.910  836.6667
## 2.915068 19386 16376.877  855.3333
## 2.917808 19387 16632.143  861.3333
## 2.920548 19388 16565.223  882.6667
## 2.923288 19389 16826.840  847.3333
## 2.926027 19390 16607.632  854.0000
## 2.928767 19391 15896.926  846.3333
## 2.931507 19392 15459.666  877.0000
## 2.934247 19393 14981.689  893.0000
## 2.936986 19394 16191.926  921.6667
## 2.939726 19395 15954.066  878.0000
## 2.942466 19396 16560.263  891.6667
## 2.945205 19397 15296.042  833.3333
## 2.947945 19398 15708.852  905.3333
## 2.950685 19399 16207.822  912.0000
## 2.953425 19400 16382.583  941.0000
## 2.956164 19401 16400.473  907.3333
## 2.958904 19402 15780.093  909.0000
## 2.961644 19403 16065.959  921.3333
## 2.964384 19404 15758.651  908.0000
## 2.967123 19405 16903.237  938.3333
## 2.969863 19406 17003.425  936.0000
## 2.972603 19407 17081.747  959.0000
## 2.975342 19408 16680.582  925.6667
## 2.978082 19409 16580.326  938.3333
## 2.980822 19410 16326.682  922.6667
## 2.983562 19411 15418.653  905.0000
## 2.986301 19412 15403.727  899.6667
## 2.989041 19413 15757.953  896.3333
## 2.991781 19414 17078.927  968.3333
## 2.994521 19415 17860.227  987.0000

plot(rollmean(dpd_ts, 3))

Mise à part l’impact qu’ont eu les données manquantes du mois d’octobre 2021 sur nos moyennes mobiles (chiffre d’affaires et nombre de ventes), nous pouvons clairement voir que ni l’une ni l’autre ne semblent connaitre de phénomène de saisonnalité. De plus, ces moyennes mobiles nous démontrent parfaitement la relative stabilité du CA au cours du temps.

Relation entre caractéristiques et comportements des clients

Relation entre genre et catégories des livres achetés

Test statistique

categ_genre <- table(data$categ, data$sex)
categ_genre

##    
##          f      m
##   0 206220 209460
##   1 114899 112270
##   2  17283  19200

Nous pouvons déjà constater qu’il y a pratiquement autant de ventes chez les hommes que chez les femmes dans chaque catégorie

library(questionr)

lprop(categ_genre)

##      
##       f     m     Total
##   0    49.6  50.4 100.0
##   1    50.6  49.4 100.0
##   2    47.4  52.6 100.0
##   All  49.8  50.2 100.0

Effectuons un test de Khi-Deux afin de vérifier quelle hypothèse nous allons accepter : Admettons en tant qu’hypothèse nulle (H0) que la catégorie des livres achetés est liée au genre du client. Admettons en tant qu’hypothèse alternative (H1) que la catégorie des livres achetés n’est pas liée au genre du client.

library(stats)

chisq.test(categ_genre)

## 
##  Pearson's Chi-squared test
## 
## data:  categ_genre
## X-squared = 147, df = 2, p-value < 2.2e-16

La p-value étant nettement inférieure à 5% ou 0.05, nous pouvons rejeter H0 et admettre que la catégorie des livres achetés n’est pas liée au genre du client.

library(questionr)

chisq.residuals(categ_genre)

##    
##         f     m
##   0 -1.86  1.85
##   1  5.16 -5.14
##   2 -6.61  6.58

Représentons graphiquement ces résultats afin de mieux les appréhender :

library(graphics)

mosaicplot(categ_genre, las = 3, shade = TRUE)

Il n’y a pas de lien établi entre le genre du client et la catégorie des livres achetés.

Détermination graphique

data_f <- data[(data$sex == "f"),]
data_m <- data[(data$sex == "m"),]
head(data_f)

##     client_id id_prod       date session_id price categ sex birth age
## 106    c_1000   1_480 2023-01-06   s_322978 19.08     1   f  1966  57
## 107    c_1000   1_267 2021-12-01   s_127695 27.99     1   f  1966  57
## 108    c_1000   1_719 2022-08-11   s_251709 41.99     1   f  1966  57
## 109    c_1000   1_378 2022-02-14   s_164815 26.61     1   f  1966  57
## 110    c_1000   1_325 2021-05-31    s_42154 27.99     1   f  1966  57
## 111    c_1000   1_441 2022-01-30   s_157167 24.99     1   f  1966  57
##     annee_mois jour mois annee
## 106 2023-01-01    6    1  2023
## 107 2021-12-01    1   12  2021
## 108 2022-08-01   11    8  2022
## 109 2022-02-01   14    2  2022
## 110 2021-05-01   31    5  2021
## 111 2022-01-01   30    1  2022

head(data_m)

##   client_id id_prod       date session_id price categ sex birth age annee_mois
## 1       c_1  0_1378 2021-08-23    s_79696 13.96     0   m  1955  68 2021-08-01
## 2       c_1  0_1090 2021-12-19   s_136532 13.78     0   m  1955  68 2021-12-01
## 3       c_1  0_1547 2021-09-08    s_86739  8.99     0   m  1955  68 2021-09-01
## 4       c_1   1_713 2021-11-15   s_120172 33.99     1   m  1955  68 2021-11-01
## 5       c_1   1_364 2022-06-15   s_224447 10.30     1   m  1955  68 2022-06-01
## 6       c_1  0_2277 2021-09-06    s_85977 10.99     0   m  1955  68 2021-09-01
##   jour mois annee
## 1   23    8  2021
## 2   19   12  2021
## 3    8    9  2021
## 4   15   11  2021
## 5   15    6  2022
## 6    6    9  2021

library(dplyr)
library(magrittr)

data_m_categ <- data_m %>% group_by(categ) %>% summarise(nb_achats = n(), prop = round(nb_achats/340930*100, digits = 2))
data_f_categ <- data_f %>% group_by(categ) %>% summarise(nb_achats = n(), prop = round(nb_achats/338402*100 , digits = 2))

library(magrittr)
library(ggplot2)
library(hrbrthemes)
library(graphics)

par(mfrow=c(2,1))

data_m_categ %>%
  ggplot(aes(x="", y=prop, fill=categ)) +
  geom_bar(stat="identity", width=1, color="white") +
  coord_polar("y", start=0) +
  theme_void() +
  ggtitle("Proportion des ventes chez les hommes, par catégorie")

data_f_categ %>%
  ggplot(aes(x="", y=prop, fill=categ)) +
  geom_bar(stat="identity", width=1, color="white") +
  coord_polar("y", start=0) +
  theme_void() +
  ggtitle("Proportion des ventes chez les femmes, par catégorie")

Il ne semble pas y avoir de lien établi entre le genre du client et les catégories des livres achetés.

Relation entre âge des clients et montant total des achats

Test statistique

Étant donné la taille de notre échantillon et d’après la loi des grands nombres, nous pouvons supposer que la moyenne empirique tend vers l’espérance et donc vers une loi normale. Nous pouvons donc effectuer un test d’anova (analyse des variances) afin de vérifier quelle hypothèse nous allons accepter : L’hypothèse nulle (H0) dit que les moyennes de tous les groupes sont égales (le montant total des achats n’est pas lié à l’âge du client). L’hypothèse alternative (H1) dit que la moyenne d’au moins l’un des groupes est différente de celles des autres (le montant total des achats est lié à l’âge du client).

library(onewaytests)

## 
## Attaching package: 'onewaytests'

## The following object is masked from 'package:questionr':
## 
##     describe

aov.test(ca~factor(age), ca_per_client)

## 
##   One-Way Analysis of Variance (alpha = 0.05) 
## ------------------------------------------------------------- 
##   data : ca and factor(age) 
## 
##   statistic  : 0.900226 
##   num df     : 75 
##   denom df   : 8524 
##   p.value    : 0.7176108 
## 
##   Result     : Difference is not statistically significant. 
## -------------------------------------------------------------

La p-value étant largement supérieure à 5% ou 0.05, nous pouvons accepter H0 et admettre que le montant total des achats n’est pas lié à l’âge du client.

Pour vérification, utilisons la corrélation des rangs de Spearman (moins sensible aux valeurs extrêmes) :

library(stats)

cor(ca_per_client$age, ca_per_client$ca, method = "spearman")

## [1] -0.1849352

Nous avons bien la confirmation qu’il n’existe pas de lien entre âge et motant total des achats dans ce cas précis (la corrélation est plus proche de 0 que de 1 ou -1).

Détermination graphique

library(dplyr)
library(magrittr)
library(ggplot2)
library(graphics)

ca_per_client %>%
  group_by(age) %>%
  summarise(ca=mean(ca)) %>%
  ggplot(aes(x=age, y=ca)) + 
  geom_bar(stat = "identity", width=0.5, color="grey", fill="lightblue") +
  xlab("Âge") +
  ylab("Montant total des achats") +
  ggtitle("Montant total des achats réalisé en fonction de l'âge (moyenne)") +
  theme_ipsum()

L’âge ne semble effectivement pas influencer le montant total des achats. En effet, bien que le montant total des achats des clients appartenant à la classe d’âge des actifs soit légèrement supérieur à celui des clients plus âgés, cette différence n’est pas suffisamment significative et nous pouvons clairement observer que les montants sont relativement bien équilibrés le long de l’axe des abscisses.

Relation entre âge des clients et taille du panier moyen

Test statistique

Effectuons de nouveau un test d’analyse des variances afin de vérifier laquelle des 2 hypothèses nous allons accepter : L’hypothèse nulle (H0) dit que les moyennes de tous les groupes sont égales (la taille du panier moyen n’est pas liée à l’âge du client). L’hypothèse alternative (H1) dit que la moyenne d’au moins l’un des groupes est différente de celles des autres (la taille du panier moyen est liée à l’âge du client).

library(onewaytests)

aov.test(panier_moyen~factor(age), ca_per_client)

## 
##   One-Way Analysis of Variance (alpha = 0.05) 
## ------------------------------------------------------------- 
##   data : panier_moyen and factor(age) 
## 
##   statistic  : 198.3447 
##   num df     : 75 
##   denom df   : 8524 
##   p.value    : 0 
## 
##   Result     : Difference is statistically significant. 
## -------------------------------------------------------------

La p-value étant nettement inférieure à 5% ou 0.05, nous pouvons rejeter H0 et admettre que la taille du panier moyen est liée à l’âge du client.

Pour vérification :

library(stats)

cor(ca_per_client$age, ca_per_client$panier_moyen)

## [1] -0.5131829

Cette fois, le coefficient de corrélation de Pearson nous indique la présence d’une faible corrélation linéaire négative (plus l’âge augmente, plus le montant du panier moyen diminue).

Détermination graphique

library(dplyr)
library(magrittr)
library(ggplot2)
library(graphics)

ca_per_client %>%
  group_by(age) %>%
  summarise(panier_moyen=mean(panier_moyen)) %>%
  ggplot(aes(x=age, y=panier_moyen)) + 
  geom_bar(stat = "identity", width=0.5, color="grey", fill="lightblue") +
  xlab("Âge") +
  ylab("Montant du panier moyen") +
  ggtitle("Taille du panier moyen en fonction de l'âge (moyenne)") +
  theme_ipsum()

Le montant du panier moyen des clients les plus jeunes (jusqu’à environ 32 ans) est beaucoup plus élevé que celui des autres clients. Il représente plus du double de celui des autres clients.

Relation entre âge des clients et fréquence d’achat

Test Statistique

library(dplyr)
library(magrittr)
library(vctrs)

## 
## Attaching package: 'vctrs'

## The following object is masked from 'package:dplyr':
## 
##     data_frame

frequence_mensuelle <- data %>% group_by(client_id, annee, mois) %>% summarise(nb_achats=vec_unique_count(session_id))

## `summarise()` has grouped output by 'client_id', 'annee'. You can override
## using the `.groups` argument.

frequence_mensuelle <- frequence_mensuelle %>% group_by(client_id) %>% summarise(freq_mensu_moyenne=nb_achats/342366)

## `summarise()` has grouped output by 'client_id'. You can override using the
## `.groups` argument.

frequence_mensuelle <- merge(frequence_mensuelle, clients2)
head(frequence_mensuelle)

##   client_id freq_mensu_moyenne sex birth age
## 1       c_1       2.920851e-06   m  1955  68
## 2       c_1       1.168340e-05   m  1955  68
## 3       c_1       2.920851e-06   m  1955  68
## 4       c_1       5.841702e-06   m  1955  68
## 5       c_1       2.920851e-06   m  1955  68
## 6       c_1       5.841702e-06   m  1955  68

library(dplyr)
library(magrittr)

frequence_age <- frequence_mensuelle %>% group_by(age) %>% summarise(freq_mensuelle=round(sum(freq_mensu_moyenne), digits = 5))
head(frequence_age)

## # A tibble: 6 × 2
##     age freq_mensuelle
##   <dbl>          <dbl>
## 1    19        0.0251 
## 2    20        0.0075 
## 3    21        0.00761
## 4    22        0.0071 
## 5    23        0.00707
## 6    24        0.0195

Effectuons un test de correlation afin de vérifier laquelle des 2 hypothèses nous allons rejeter : Admettons en tant qu’hypothèse nulle (H0) que la fréquence des achats n’est pas liée à l’âge du client. Admettons en tant qu’hypothèse alternative (H1) que la fréquence des achats est liée à l’âge du client.

library(stats)

cor.test(frequence_age$age, frequence_age$freq_mensuelle, method=c("pearson", "kendall", "spearman"), alternative="two.sided", conf.level=0.95)

## 
##  Pearson's product-moment correlation
## 
## data:  frequence_age$age and frequence_age$freq_mensuelle
## t = -5.7072, df = 74, p-value = 2.238e-07
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  -0.6920393 -0.3740004
## sample estimates:
##        cor 
## -0.5528402

La p-value étant nettement inférieure à 5% ou 0.05, nous pouvons rejeter H0 et admettre que la fréquence des achats est liée à l’âge du client.

Détermination graphique

plot(frequence_age$age, frequence_age$freq_mensuelle, xlab="Âge", ylab="Fréquence des achats", main="Fréquence mensuelle d'achat par âge")

Au vu des des précédents graphiques, nous pouvons admettre qu’il existe effectivement une corrélation entre fréquence d’achat et âge. En effet nous avons pu constater que les clients les plus jeunes (les moins de 20 ans) et les clients les plus âgés (les plus de 60 ans) commandent moins et donc moins fréquemment que les clients appartenant aux catégories intermédiaires (entre 20 et 59 ans).

Relation entre âge des clients et catégories des livres achetés

Test statistique

Effectuons un test de Kruskal-Wallis (test non paramétrique, n’induisant aucune loi et permettant de comparer 2 échantillons ou plus) afin de vérifier laquelle des 2 hypothèses nous allons rejeter : L’hypothèse nulle (H0) dit que les paramètres de la distribution sont les mêmes dans chaque groupe (la catégorie des livres achetés n’est pas liée à l’âge du client). L’hypothèse alternative (H1) dit que les paramètres de la distribution diffèrent dans au moins l’un des groupes (la catégorie des livres achetés est liée à l’âge du client dans au moins l’un des cas).

library(stats)

kruskal.test(data$age, data$categ)

## 
##  Kruskal-Wallis rank sum test
## 
## data:  data$age and data$categ
## Kruskal-Wallis chi-squared = 79351, df = 2, p-value < 2.2e-16

La p-value étant nettement inférieure à 5% ou 0.05, nous pouvons rejeter H0 et admettre que la catégorie des livres achetés est liée à l’âge du client.

Détermination graphique

library(ggplot2)
library(viridis)

## Loading required package: viridisLite

library(hrbrthemes)
 
data %>%
  ggplot(aes(x=categ, y=age, fill=categ)) +
    geom_boxplot() +
    scale_fill_viridis(discrete = TRUE, alpha=0.6, option="A") +
    theme_ipsum() +
    theme(legend.position="none", plot.title = element_text(size=11)) +
    ggtitle("Répartition de l'âge des clients en fonction des catégories achetées") +
    xlab("")

categ <- rep(c("0" , "1" , "2"), 4)
tranche_age <- c(rep("Moins de 20 ans", 3), rep("20 à 39 ans", 3), rep("40 à 59 ans", 3), rep("60 ans et plus", 3))
nb_ventes <- c(3418, 5717, 6012, 131025, 60193, 27747, 239419, 107710, 1825, 41818, 53547, 899)
proportion_ventes <- c(22.56, 37.75, 39.69, 59.84, 27.49, 12.67, 68.61, 30.87, 0.52, 43.44, 55.63, 0.93)

data_categ_age <- data.frame(categ, tranche_age, nb_ventes, proportion_ventes)

library(dplyr)
library(magrittr)
library(ggplot2)
library(viridis)

data_categ_age %>%
  ggplot(aes(fill=categ, y=proportion_ventes, x=tranche_age)) +
    geom_bar(position="fill", stat="identity") +
    scale_fill_viridis(discrete = T, option = "G") +
    xlab("Tranche d'âge") +
    ylab("Proportion des ventes") +
    ggtitle("Répartition des ventes par tranche d'âge") +
    theme_ipsum()

Les moins de 20 ans achètent majoritairement des livres correspondant à la catégorie 2. Nous pouvons également constater que plus l’âge augmente, moins la catégorie 2 est achetée.

library(dplyr)
library(magrittr)
library(ggplot2)
library(viridis)

data_categ_age %>%
  ggplot(aes(fill=tranche_age, y=nb_ventes, x=categ)) +
    geom_bar(position="fill", stat="identity") +
    scale_fill_viridis(discrete = T) +
    xlab("Catégories") +
    ylab("Proportion des ventes") +
    ggtitle("Répartition des ventes par catégorie") +
    theme_ipsum()

Ce graphique nous montre, quant à lui, que les clients âgés de moins de 40 ans sont les principaux consommateurs de la catégorie 2 avec près de 90% des ventes réalisées (les adultes entre 20 et 39 ans représentants à eux seuls environ 75% des acheteurs).

ARIMA

L’analyse des moyennes mobiles nous a permis d’affirmer que ni les ventes ni le chiffre d’affaires ne connaissent de phénomène de saisonnalité. Nous pouvons donc utiliser ARIMA afin de poursuivre notre analyse.

library(tsibble)

dpm2 <- dpm
dpm2$annee_mois <- yearmonth(dpm2$annee_mois)

Nous devons d’abord choisir les ordres p, d et q de notre modèle. Nous commençons par vérifier si la série doit être différenciée pour obtenir une série stationnaire. La fonction unitroot_ndiffs effectue un test statistique pour déterminer le nombre minimum de différenciations à réaliser.

library(feasts)

## Loading required package: fabletools

## 
## Attaching package: 'fabletools'

## The following objects are masked from 'package:DescTools':
## 
##     MAE, MAPE, MSE, RMSE

library(fabletools)

unitroot_ndiffs(dpm2$ca)

## ndiffs 
##      0

Ici, le résultat est 0 donc aucune différenciation n’est nécessaire (d=0).

library(feasts)
library(fabletools)

dpm_tsb <- as_tsibble(dpm2)

## Using `annee_mois` as index variable.

library(stats)

acf(dpm_tsb$ca, lag.max = 12, type = "correlation")

On voit que le CA d’un mois m n’influe pas sur le CA des mois suivants. Effectuons maintenant un test de stationarité (test de Dickey-Fuller) :

library(tseries)

## Registered S3 method overwritten by 'quantmod':
##   method            from
##   as.zoo.data.frame zoo

adf.test(dpm_tsb$ca)

## 
##  Augmented Dickey-Fuller Test
## 
## data:  dpm_tsb$ca
## Dickey-Fuller = -2.5404, Lag order = 2, p-value = 0.3665
## alternative hypothesis: stationary

L’hypothèse H0 est que la série comporte une racine unitaire (CA non stationnaire). L’hypothèse H1 est que la série ne comporte pas de racine unitaire (CA stationnaire). La p-value étant supérieure à 0,05, nous allons accepter H0 et admettre que le CA n’est pas stationnaire. Hors, la stationnarité étant une condition nécessaire pour représenter une série temporelle avec un modèle ARIMA, créons une série différenciée sur laquelle nous effectuerons un nouveau test de stationnarité.

library(tseries)

diff_ca <- dpm_tsb$ca[2:24]-dpm_tsb$ca[1:23]
adf.test(diff_ca)

## 
##  Augmented Dickey-Fuller Test
## 
## data:  diff_ca
## Dickey-Fuller = -3.5499, Lag order = 2, p-value = 0.05696
## alternative hypothesis: stationary

Au vue de la valeur de la p-value, nous prendrons le parti d’accepter l’hypothèse H1 disant que la série ne comporte pas de racine unitaire et que, par conséquent, le CA est stationnaire. Nous pouvons donc poursuivre en réalisant nos prévisions sur diff_ca. Réalisons un nouveau test d’auto-corrélation :

library(tseries)

acf(diff_ca, lag.max = 12, type = "correlation")

Sur diff_ca, nous pouvons observer un retard de 1. Voyons maintenant quel modèle ARIMA nous allons utiliser grâce à la fonction auto.arima qui va déterminer pour nous les valeurs de p, d et q :

library(forecast)

## 
## Attaching package: 'forecast'

## The following object is masked from 'package:DescTools':
## 
##     BoxCox

auto.arima(diff_ca)

## Series: diff_ca 
## ARIMA(0,0,1) with zero mean 
## 
## Coefficients:
##           ma1
##       -0.9527
## s.e.   0.2572
## 
## sigma^2 = 1.846e+09:  log likelihood = -278.63
## AIC=561.26   AICc=561.86   BIC=563.53

library(stats)

arima(diff_ca, order=c(0,0,1))

## 
## Call:
## arima(x = diff_ca, order = c(0, 0, 1))
## 
## Coefficients:
##           ma1  intercept
##       -0.9997   1194.649
## s.e.   0.1303   1189.847
## 
## sigma^2 estimated as 1.628e+09:  log likelihood = -278.14,  aic = 562.28

library(stats)

residuals(arima(diff_ca, order=c(0,0,1)))

## Time Series:
## Start = 1 
## End = 23 
## Frequency = 1 
##  [1]   -5281.096    9665.294   -1880.424   -3677.803   -4542.845   18195.565
##  [7] -159844.967   42163.788   45796.321   39711.229   44973.161   20892.158
## [13]   -3419.478   19004.275   -3931.693    9533.076    3586.114   -9784.770
## [19]    3020.221   -9265.319    3186.371    9048.911  -52095.550

library(stats)

predict(arima(diff_ca, order=c(0,0,1)), n.ahead = 3)

## $pred
## Time Series:
## Start = 24 
## End = 26 
## Frequency = 1 
## [1] 52184.591  1194.649  1194.649
## 
## $se
## Time Series:
## Start = 24 
## End = 26 
## Frequency = 1 
## [1] 41171.49 57048.87 57048.87

library(forecast)

forecast(arima(diff_ca, order=c(0,0,1)), h=3)

##    Point Forecast       Lo 80     Hi 80      Lo 95    Hi 95
## 24      52184.591   -578.7982 104947.98  -28510.05 132879.2
## 25       1194.649 -71916.4149  74305.71 -110619.07 113008.4
## 26       1194.649 -71916.4149  74305.71 -110619.07 113008.4

autoplot(forecast(arima(diff_ca, order=c(0,0,1)), h=3))